Third order wave force on axisymmetric bodies

The third order triple-frequency wave load on fixed axisymmetric bodies by monochromatic waves is considered within the frame of potential theory. Waves are assumed to be weak non-linearity and a perturbation method is used to expand velocity potentials and wave loadings into series according to a wave steepness of kA. Integral equation method is used to compute velocity potentials up to second order in wave steepness. The third order triple-frequency wave loads are computed by an indirect method and an efficient method is applied to form the third order forcing term on the free surface quickly. The method can be used to compute third order triple-frequency surge force, heave force and pitch moment on any revolution bodies with vertical axes. The comparison with Malenica and Molin's results is made on surge force on a uniform cylinder, and comparison with experimental results is made on third order surge force, heave force and pitch moment on a truncated cylinder. More numerical computations are carried out for third order forces and moments on a uniform cylinder, truncated cylinders and a hemisphere.     

轴对称物体上的三阶波浪力

对于轴对称物体,提出了一个三阶波浪力的全绕射计算方法,自由水面上的三阶强迫项采用向外递推的方法加以计算。本方法已在计算机上实现。对于均匀圆柱问题,本方法计算结果与Malenica的半解析解吻合良好。本方法还被用于计算圆柱上的三阶波浪力矩,结果发现在低频区三阶波浪力矩具有很大的量值。     

Linear and nonlinear interactions of surface waves with bodies by a three-dimensional Rankine panel method

Linear and second-order surface wave interactions with floating and bottom-mounted bodies of realistic geometry are simulated in the time domain by a three-dimensional Rankine panel method. The fundamental stability analysis governing the propagation of transient wave disturbances on a panel mesh distributed on the free surface is carried out from first principles. The radiation condition is enforced by a dissipative beach selected to coincide with an outer annulus of panels. The fundamental physics governing the wave energy absorption is presented and the beach attributes are selected and validated for the linear and second-order problems. Computations are presented of the linear and sum-frequency second-order forces on a single and multiple truncated circular cylinders, and very good agreement is found with benchmark computations. The accuracy and efficiency of this method render it a promising candidate for the study of complex nonlinear wave induced phenomena upon offshore platforms, like springing and ringing.     

A comparison of complete and approximate solutions for second-order diffraction loads on arrays of vertical ci cular cylinders

This paper presents a comparison between two theoretical methods for computing the second-order diffraction loads on arrays of bottom-mounted, surface-piercing vertical circular cylinders in regular waves. One method presents a complete solution for the second-order hydrodynamic loads on the cylinder array via a numerical integration over the mean fluid free-surface. The other method is based on a large spacing approximation between the array members and involves the solution of a set of equivalent isolated body problems to obtain estimates for the second-order hydrodynamic loads. Numerical results for a pair of cylinders indicate very good agreement between the two methods at center-to-center spacing of both three and five radii, indicating that the approximate method may be sufficient to compute hydrodynamic interference effects to the second-order in many practical engineering situations.     

Wave drift enhancement effects in multi column structures

A theoretical assessment is made of mean wave drift forces on groups of vertical circular cylinders, such as the columns of a floating offshore platform. A complete analytical solution is obtained for two cylinders extending from seabed to free surface, and a long wave approximation is found to provide reliable predictions of the drift force in line with the waves at low frequencies. For moderate separation between the two cylinders, this force is found to tend at low frequencies to a value four times the force on an isolated cylinder.A numerical method is employed to study two surface piercing cylinders truncated below the free surface, and an arrangement of four vertical cylinders characteristic of a floating offshore platform. The mean vertical drift force is found to be reasonably well approximated, over the frequency range of practical interest, by the force on an individual cylinder considered in isolation multiplied by the number of cylinders in the group. Interaction effects, however, have a profound influence on the total horizontal drift force. At low frequencies this force is found to tend to the force on an isolated cylinder multiplied by the squate of the number of cylinders in the group.     

Second-Order Wave Diffraction Around 3-D Bodies by A Time-Domain Method

A time-domain method is applied to simulate nonlinear wave diffraction around a surface piercing 3-D arbitrary body. The method involves the application of Taylor series expansions and the use of perturbation procedure to establish the corresponding boundary value problems with respect to a time-independent fluid domain. A boundary element method based on B-spline expansion is used to calculate the wave field at each time step, and the free surface boundary condition is satisfied to the second order of wave steepness by a numerical integration in time. An artificial damping layer is adopted on the free surface for the removal of wave reflection from the outer boundary. As an illustration, the method is used to compute the second-order wave forces and run-up on a surface-piercing circular cylinder. The present method is found to be accurate, computationally efficient, and numerically stable.     

A novel decomposition of the quadratic transfer function (QTF) for the time-domain simulation of non-linear wave forces on floating bodies

In the present study, a novel method is proposed for the separation of the second-order sum- and difference-frequency wave forces—that is, quadratic transfer functions (QTFs)—on a floating body into three components due to wave–wave, wave–motion, and motion–motion action. By applying the new QTF components, the second-order wave forces on a floating body can be strictly computed in the time domain. In this work, the boundary value problems (BVPs) corresponding to the three kinds of QTF components were derived, and non-homogeneous boundary conditions on the free surface and the body surface were obtained. The second-order diffraction potentials were determined using the boundary integral equation method. In the solution procedure, the highly oscillatory and slowly converging integral on the free surface was evaluated in an accurate and effective manner. Furthermore, the application of the QTF components in the time domain was demonstrated. The second-order exciting forces in the time domain were divided into three parts. Each part of these forces was computed via a two-term Volterra series model based on the incident waves, the first-order motion response, and the QTF components. This method was applied to several numerical examples. The results demonstrated that this decomposition yields satisfactory results.     

Fully nonlinear analysis of near-trapping phenomenon around an array of cylinders

The wave diffraction around an array of fixed vertical circular cylinders is simulated in a numerical wave tank by using a fully nonlinear model in the time domain. The emphasis of the paper lies in the insightful investigation of the nonlinear properties of the near-trapping phenomenon associated with the multiple cylinders. The numerical model is validated by analytical solutions as well as experimental data for waves propagating past two and four vertical cylinders in certain arrangements. An array of four identical circular cylinders at the corners of a square with an incident wave along the diagonal of the square is the main focus here for investigating the near-trapping phenomenon. When near-trapping occurs, the present study shows that an extremely high wave elevation near the cylinders can be observed. At the same time, the hydrodynamic forces on different cylinders are found to be either in phase or out of phase, leading to some characteristic force patterns acting on the whole structure. Due to the nature of the numerical model adopted, nonlinearity at different orders can be captured using a harmonic analysis. In addition to first- and second-order near-trapping, the third-order (triple-frequency) nonlinear component is presented for the first time. For the configuration selected, it is found that at one specific incident wave frequency and direction one trapped mode is excited by second-order effects, while a different trapped mode (having similar symmetries) is excited by the third harmonic of the incident wave frequency.     

波浪与结构物作用分析的一种高阶边界元方法--自由项和柯西主值积分的直接数值计算

采用直接数值计算方法计算了势流问题高阶边界元方法中的自由项系数和柯西主值积分,建立了波浪与结构物作用的一种高阶边界元方法.通过算例研究了物体表面上固角系数的计算精度和不同网格剖分、不同阶高斯积分点对柯西主值积分的影响.对截断圆柱上的波浪作用力与解析解做了对比,发现本方法具有很高的计算精度,随网格的加密迅速收敛于解析解.     

Time domain simulation of side-by-side floating bodies using a 3D numerical wave tank approach

The coupled system of two side-by-side fixed and/or floating bodies interacting with a large amplitude nonlinear wave is studied using a direct time domain solution method. The numerical model is based on a three-dimensional mixed Eulerian–Lagrangian (MEL) method under certain simplifying approximations permitting Rankine panel scheme to be implemented over a time-invariant boundary surface to solve the boundary value problem for the unknown velocity potentials. A 4th order Adams–Bashforth–Moulton scheme is used for time marching of rigid-body motion histories of the individual bodies and evolution of the free-surface including the gap region in which large resonant fluid motions occur. A systematic study has been carried out to evaluate the performance of the developed time domain method in simulating the forces and motions as well as the fluid motion in the gap region for the two body system under various arrangements and in different wave-headings. At first, the computed numerical results have been validated and verified with computational and experimental results available in literature for standard geometries such as vertical truncated cylinders and rectangular boxes. Secondly, effectiveness of the damping lid model which is introduced to suppress wave resonance in the gap region is investigated including its influence on maximum sway forces on fixed and floating rectangular barges in side-by-side configurations. Thirdly, comparative studies on absolute and relative motion response for two cases (two rectangular barges, and a FLNG-FPSO + shuttle tanker) in side-by-side arrangement are detailed to bring out the importance of nonlinearities arising due to steep nonlinear incident waves. Finally, coupled motions of the two-body system of an FPSO and a shuttle tanker floating in side-by-side configuration in a steep nonlinear wave field are studied in which the two bodies are connected through hawsers, and also the FPSO is moored to the ground. Additionally there is a fender between the two bodies.     

双色入射波下二阶波浪力响应函数

应用边界元方法,对双频入射波在和频及差频下的二阶速度势做了完整的求解,通过物面积分计算了任意三维结构上的二阶波浪力的传递函数.对简单几何物体,与发表的结果做了对比,两者吻合良好,验证了本方法的正确性.应用这一方法还对复杂的张力腿平台模型做了实际计算,发现在低频和高频区二阶波浪力平方响应函数有着显著的能量分布.     

Convergence criteria for axisymmetric Green's function with application to floating bodies

Convergence criteria are provided for truncating the evanescent eigenmode series in the Green's function for vertical, axisymmetric bodies of revolution. To numerically compute the strength of the source distribution for both the exciting and restoring forces, separate criteria are required for the off-diagonal and for the diagonal elements in the matrix of coefficients for the source strengths. The Black-Fenton algorithms for wave-induced exciting forces on vertical axisymmetric bodies of revolution are extended to include the wave-induced restoring forces. The Gauss-Seidel matrix version method recommended by Fenton is compared with the Gauss elimination method and is found to be non-converging for near deep water wave conditions. Comparison between theory and measured data for the dynamic response of a discus buoy demonstrates the convergence criteria over a range of dimensionless frequencies in relatively deep water wave conditions.     

三维物体上二阶波浪绕射的实时模拟

采用二阶时域理论对非线性波浪在任意三维物体周围的绕射问题进行了研究，对自由表面边界条件进行Taylor级数展开，应用摄动展开可以建立相应的边值问题，而且此边值问题的计算域不随时间变化，运用基于B－样条的边界元方法求解每一时刻的波浪场，二阶自由表面边界条件在时间上进行数值积分，在自由表面加了一个人工阻尼层以避免波浪的反射，速度势分解为已知的入射势和未知的散射势，初始条件采用二阶Stokes波浪场，通过加入物体表面边界条件，得到散射势在时间和空间上的发展，本文对圆柱所受规则波的二阶波浪力和波浪爬高进行了计算，数值结果表明此理论计算准确，效率高，数值稳定。     

Second-Order Potentials and Forces for Two-Dimensional Diffraction Problem of Finite Water Depth

-In this paper, an analytical solution in the outer region of finite water depth is derived for the second-order diffraction potential, which gives a clear physical meaning of the wave transmission and reflection characteristics in the far field. A numerical method-simple Green's function technique-for calculating the second-order diffraction potential in the inner region is also described. Numerical results are provided for the second-order wave forces on a semi-submerged cylinder. It is found that the contribution of second-order diffraction potential to second-order wave forces is important. The effect of water depth and submerged depth on the wave force is also discussed.     

Wave enhancement due to blockage in semi-submersible and TLP structures

An exact analytical method is described to solve the diffraction problem of a group of truncated vertical cylinders. In order to account for the interaction between the cylinders, Kagemoto and Yue's exact algebraic method is utilised (Kagemoto, H., Yue, D.K.P., 1986. Interactions among multiple three-dimensional bodies in water waves: an exact algebraic method. J Fluid Mech, 46, 129–139). The isolated cylinder diffraction potential is obtained using Garret's solution and evanescent mode solutions are derived in a similar manner (Garret, C.J.R., 1971. Wave forces on a circular dock. J Fluid Mech, 46, 129–139).Free surface elevations are calculated for an array of four cylinders and compared with experiments. Comparisons show good agreement.     

Interactions between nonlinear water waves and non-wall-sided 3D structures

Interactions between water waves and non-wall-sided cylinders are analyzed based on velocity potential theory with fully nonlinear boundary conditions on the free surface and the body surface. The finite element method (FEM) is adopted together with a 3D mesh generated through an extension of a 2D Delaunay grid on a horizontal plane along the depth. The linear matrix equation for the velocity potential is constructed by imposing the governing equation and boundary conditions through the Galerkin method and is solved through an iterative method. By imposing the gradient of the potential equal to the velocity, the Galerkin method is used again to obtain the velocity field in the fluid domain. Simulations are made for bottom mounted and truncated cylinders with flare in a numerical tank. Periodic waves and wave groups are generated by a piston type wave maker mounted on one end of the tank. Results are obtained for forces, wave profiles and wave runups. Further simulations are made for a cylinder with flare subjected to forced motion in otherwise still open water. Results are provided for surge and heave motion in different amplitudes, and for a body moving in a circular path in the horizontal plane. Comparisons are made in several cases with the results obtained from the second order solution in the time domain.     

Benchmark computations of wave run-up on single cylinder and four cylinders by naoe-FOAM-SJTU solver

The benchmark simulations of wave run-up on a fixed single truncated circular cylinder and four circular cylinders are presented in this paper. Our in-house CFD solver naoe-FOAM-SJTU is adopted which is an unsteady two-phase CFD code based on the open source package OpenFOAM. The Navier-Stokes equations are employed as the governing equations, and the volume of fluid (VOF) method is applied for capturing the free surface. Monochromatic incident waves with the specified wave period and wave height are simulated and wave run-up heights around the cylinder are computed and recorded with numerical virtual wave probes. The relationship between the wave run-up heights and the incident wave parameters are analyzed. The numerical results indicate that the presented naoe-FOAM-SJTU solver can provide accurate predictions for the wave run-up on one fixed cylinder and four cylinders, which has been proved by the comparison of simulated results with experimental data.     

层化海洋中二阶多色波对可渗透结构的绕射问题

可渗透结构具有使波浪作用减弱的效应,而海水的层化及水波的非线性使结构的波绕射产生多层复杂机制。将可渗透结构应用于复杂海况条件中,海水的层化性、波浪的非线性及结构的透空性构成了波绕射的一个十分复杂的数学问题。该问题存在理论研究的必要性,而文章则着重探讨其数学分析的可能性。通过引入二层海的层化海模式及Stokes二阶波的非线性波模式,给出了二阶多色波对透空结构的波绕射的定解问题提法,提出了复合形式的二阶多色波辐射条件式及可渗透结构的二阶物面条件式,应用特征函数解法与积分法推导了多色波对结构绕射的一阶势解与二阶作用的耦合积分解式,并讨论了解式所涉及无穷积分的算法。     

Fully nonlinear simulations of wave resonance by an array of cylinders in vertical motions

The finite element method(FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions.The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions.The finite element linear system is solved by the conjugate gradient(CG) method with a symmetric successive overelaxlation(SSOR) preconditioner.The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation.Numerical examples are given by an array of floating wedgeshaped cylinders and rectangular cylinders.Results are provided for heave motions including wave elevations,profiles and hydrodynamic forces.Comparisons are made in several cases with the results obtained from the second order solution in the time domain.It is found that the wave amplitude in the middle region of the array is larger than those in other places,and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.