Assessing Lagrangian schemes for simulating diffusion on non-flat isopycnal surfaces

In large-scale ocean flows diffusion mostly occurs along the density surfaces and its representation resorts to the Redi isopycnal diffusivity tensor containing off-diagonal terms. This study focuses on the Lagrangian/particle framework for simulating such diffusive processes. A two-dimensional idealised test case for purely isopycnal diffusion on non-flat isopycnal surfaces is considered. Implementation of the higher order strong Euler, Milstein and order 1.5 Taylor schemes on our idealised test case shows that the higher order strong schemes produce the better pathwise approximations. The effective spurious diapycnal diffusivity is measured for each Lagrangian scheme under consideration. The propensity of the particles to move away from the isopycnal surface on which they were released is also measured. This shows that for non-flat isopycnals the order of convergence of the Euler scheme is not sufficient to achieve the desired accuracy. However, the Milstein scheme seems to be a good choice to achieve in an efficient way a fairly accurate result.     

Application of particle filter combined with extended Kalman filter in model identification of an autonomous underwater vehicle based on experimental data

This study proposes a method for identification of the nonlinear dynamic model of an AUV while some states are unmeasured; hence, it concentrates on a nonlinear “state and parameter estimation” issue. In this method, a local linearization is used for solving the nonlinear dynamics based on the extended Kalman filter (EKF), and a particle filter (PF) is used to minimize errors and variances of the nonlinear system. In other words, the PF is combined with the EKF in the form of the extended Kalman particle filter (EKPF). The EKPF method is independent of the initial values and satisfies the limits of the parameters and also the assumption that the hydrodynamic coefficients are constant. Hence, it is shown when the ranges or signs of some parameters are known, the EKPF is a more accurate estimator than the EKF. Moreover, a new simulation is done using the model estimated by the EKPF and the results are compared and validated with the measured data of a new experimental test. It is shown that the obtained model can predict the trajectory path with the total normalized root-mean-square error (NRMSE) of 14% and the surge mean speed with the NRMSE of 5%; and it describes the 6DOF motion of the AUV more accurate than the EKF model.     

A method for second-order diffraction potential from an axisymmetric body

The paper provides a detailed analysis for the second-order diffraction of monochromatic waves. For the second-order potential on the free surface, the paper proposed a forward prediction method for computing the integration on the free surface. By this method we only need to run the infinity integration on the free surface directly for a few points; a one-step quadrature can then be applied successively outward from the body for potentials at other points. For wave diffraction from a body of revolution with a vertical axis, the paper derives a new integral equation, which can cancel the leading singularity in the derivative of ring Green's functions automatically. To obtain accurate results, different approaches are also used to deal with singularities in the ring Green's functions in the integration on both the body surface and free surface. The method has been implemented for bodies of revolution with vertical axes, but the theory is also available for arbitrary bodies.A numerical examination is made to validate the numerical code by comparing the second-order forces and moments on uniform and truncated cylinders and second-order diffraction potentials on the free surface with some published results. The comparison shows that the present results are in good agreement with those published. The method is also used to compute the second-order wave elevation around uniform and truncated cylinders.     

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。     

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.     

A numerical study of second-order springing of an elastic body using higher-order boundary element method (HOBEM)

Second-order springing on an elastic body with forward speed is analyzed by numerical simulations. The boundary-value problem for the velocity potential is solved by means of the direct time-domain higher-order boundary element method (HOBEM). The free-surface boundary condition in the boundary-value problem is approximated on the mean surface up to second order by use of perturbation and Taylor-series expansion methods. The body boundary condition for an elastic body is derived with various quantities which are redefined in the generalized mode. These variables such as mode shape, normal vector, etc. are obtained by using directional derivative and continuum mechanics, and the same mathematical expressions are used to obtain several second-order generalized forces. To validate the numerical results, the second-order hydrodynamic force on the bottom-mounted rigid/elastic cylinders without forward speed is compared with other semi-analytic results. The property of second-order forces on an elastic ship is studied by changing the flexural rigidity and forward speed with elastic response. It is confirmed that the second-order velocity potential is important for a body with forward speed and investigation should be made more on numerical methods for accurate computation of the second-order velocity-potential force with forward speed.     

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.     

Elastic lateral-torsional buckling analysis of permanent means of access structure

International Maritime Organization (IMO) adopted a new regulation to request permanent means of access (PMA) for a regular inspection of ship structure. Horizontal platforms for an inspector to walk on should be provided at specified locations. The platform is attached perpendicular to longitudinal bulkheads or side shell like a common longitudinal stiffener. Since the platform is much wider than ordinary stiffeners, a mid-flat-bar is welded in the middle of the platform. The wide platform (i.e. the tall web plate) makes PMA structure prone to lateral torsional buckling prior to overall flexural Euler buckling subjected to axial compression. This study employs the Rayleigh-Ritz method to treat the elastic lateral-torsional buckling of the PMA structure. The deformation of the cross-section can be expressed using six independant parameters. Compared to the previous research for an ordinary stiffened plate (Hughes and Ma, 1996a), two additional parameters are employed to model the deformation of the mid flat bar. This study also proposes a new strain distribution of lateral bending introducing two respective neutral axes for the flange and the mid-flat-bar. Two mathematical models are developed for two cases; one without associated plating, and the other with both the plating and its rotational restraint. In the former, the coupling between the lateral torsional buckling (“tripping”) and the Euler buckling is investigated. In the latter, a plate rotational spring constant is suggested based on extended deformed shape of the plating. For each model, the validity of the proposed method is verified by a comparison with a number of linear buckling analyses carried out using the NASTRAN finite-element program.     

Enhanced predictions of wave impact pressure by improved incompressible SPH methods

A new criterion is proposed for a more efficient assessment of free-surface particles in a particle-based simulation. Enhanced wave impact simulations are carried out by improved Incompressible SPH (ISPH) methods. The first improvement is the same as that in the Corrected ISPH (CISPH; [Khayyer A, Gotoh, H, Shao SD. Corrected incompressible SPH method for accurate water-surface tracking in breaking waves, Coast Eng 2008; 55 (3): 236–250]) method and is proposed for the improvement of momentum conservation. The second improvement is achieved by deriving and employing a higher order source term based on a more accurate differentiation to obtain a less fluctuating and more accurate pressure field. The enhanced performance of improved ISPH methods is demonstrated through the simulation of several fluid impact simulations in comparison with the experimental data and simulation results by other numerical methods.     

Surface gravity waves from direct numerical simulations of the Euler equations: A comparison with second-order theory

When the wave spectrum is sufficiently narrow-banded and the wave steepness is sufficiently high, the modulational instability can take place and waves can be higher than expected from second-order wave theory. In order to investigate these effects on the statistical distribution of long-crested, deep water waves, direct numerical simulations of the Euler equations have been performed. Results show that, for a typical design spectral shape, both the upper and lower tails of the probability density function for the surface elevation significantly deviate from the commonly used second-order wave theory. In this respect, the crest elevation is observed to increase up to 18% at low probability levels. It would furthermore be expected that wave troughs become shallower due to nonlinear effects. Nonetheless, the numerical simulations show that the trough depressions tend to be deeper than in second-order theory.     

Improved numerical solution of Dobrovol'skaya's boundary integral equations on similarity flow for uniform symmetrical entry of wedges

Dobrovol'skaya [1] presented a similarity solution for the water entry of symmetrical wedges with constant velocity. The solution involves an integral equation that becomes increasingly harder to numerically solve as the deadrise angle decreases. Zhao and Faltinsen [2] were able to present reliable results for deadrise angles down to 4°. In this paper, Zhao and Faltinsen's results are improved and reliable results for deadrise angles down to 1° are confirmed by comparing to the asymptotic solutions at small deadrise angles and the solutions by the traditional boundary element method at relatively large deadrise angles. The present similarity solution results provide a reference solution in theoretical studies of water entry problems and in developing accurate numerical solvers for simulating strongly nonlinear wave–body interactions, which flows are governed by Laplace equation or Euler equation.     

基于拉格朗日拟序结构的黑潮区域海流运输结构提取和分析

海流的拉格朗日运动对于研究物质输送有着重要意义,拉格朗日拟序结构(LCSs)作为研究海流结构的新型方法,相比于传统欧拉方法更为客观。本文提出了一种新的计算LCSs束的方法,基于25年的平均速度场,利用变分方法计算得到黑潮区域的气候态LCSs,并通过简化合并的方法得到了气候态LCSs束,该LCSs束能够突出地显示出海流特性和运输模式,其代表的平均拉格朗日环流有很强的约束作用,且具有鲁棒性。最终我们获得了气候态下12个月份的流场结构图,揭示了月周期性拉格朗日环流规律。本文还利用虚拟粒子输运、多年浮标轨迹以及气候态温盐异常3种方法进行了验证,与拉格朗日运输模式相吻合,证明了海流拉格朗日拟序结构的准确性和可靠性。     

A Finite-Difference Approach to the Time-Dependent Mild-Slope Equation

A finite-difference approach is used to develop a time-dependent mild-slope equation incorporating the effects of bottom dissipation and nonlinearity.The Euler predictor-corrector method and the three-point finite-difference method with varying spatial steps are adopted to discretize the time derivatives and the two-dimensional horizontal ones,respectively,thus leading both the time and spatial derivatives to the second-order accuracy.The boundary conditions for the present model are treated on the basis of the general conditions for open and fixed boundaries with an arbitrary reflection coefficient and phase shift.Both the linear and nonlinear versions of the numerical model are applied to the wave propagation and transformation over an elliptic shoal on a sloping beach,respectively,and the linear version is applied to the simulation of wave propagation in a fully open rectangular harbor.From comparison of numerical results with theoretical or experimental ones,it is found that they are in reasonable agreement.     

Wave crest and trough distributions in a broad-banded directional wave field

It is well established that the modulational instability enhances the probability of occurrence for extreme events in long crested wave fields. Recent studies, however, have shown that the coexistence of directional wave components can reduce the effects related to the modulational instability. Here, numerical simulations of the Euler equations are used to investigate whether the modulational instability may produce significant deviations from second-order statistical properties of surface gravity waves when short crestness (i.e., directionality) is accounted for. The case of a broad-banded directional wave field (i.e. wind sea) is investigated. The analysis is concentrated on the wave crest and trough distribution. For completeness a comparison with a unidirectional wave field is presented also. Results will show that the distributions based on second-order theory provide a good estimate for the simulated crest and trough height also at low probability levels.     

胶州湾潮余流和物质输送之间的关系

基于MIKE21的HD模型,通过模拟得到胶州湾的潮流场,胶州湾在涨急时最大流速1.04m/s,落急时最大流速约为0.96m/s。胶州湾余流总体较小,平均为0.03m/s左右。并在湾内不同位置释放自由粒子,以MIKE21的Particle tracking模型计算出其在潮流作用下的运移轨迹,结果表明粒子大多数运移到湾内近岸区域,少部分在湾口区域附近;在潮流场基础上计算了欧拉余流场,并和粒子运移结果进行对比,表明欧拉余流场在区域流向比较一致时可以表示粒子运动的趋势,为物质迁移、控制污染等方面提供了一定的参考借鉴意义。     

等幅疲劳三参数P—S—N曲线族最优拟合法

对于给定的等幅疲劳成组试验数据 ,利用三参数幂函数方程来描述P -S -N等幅曲线。通过线性拟合相关系数的最优化来确定其中三个系数 ,采用割线法进行最优化求解。该方法具有迭代收敛快、拟合精度高的特点     

A Eulerian–Lagrangian method for the simulation of wave propagation

A Eulerian–Lagrangian method (ELM) is employed for the simulation of wave propagation in the present research. The wave action conservation equation, instead of the wave energy balance equation, is used. The wave action is conservative and the action flux remains constant along the wave rays. The ELM correctly accounts for this physical characteristic of wave propagation and integrates the wave action spectrum along the wave rays. Thus, the total derivative for wave action spectrum may be introduced into the numerical scheme and the complicated partial differential wave action balance equation is simplified into an ordinary differential equation. A number of test cases on wave propagation are carried out and show that the present method is stable, accurate and efficient. The results are compared with analytical solutions and/or other computed results. It is shown that the ELM is superior to the first-order upwind method in accuracy, stability and efficiency and may better reflect the complicated dynamics due to the complicated bathymetry features in shallow water areas.     

A two-dimensional numerical wave flume based on SA-MPLS method

A spatially adaptive(SA) two-dimensional(2-D) numerical wave flume is presented based on the quadtree mesh system,in which a new multiple particle level set(MPLS) method is proposed to solve the problem of interface tracking,in which common intersection may be traversed by multiple interfaces.By using the adaptive mesh technique and the MPLS method,mesh resolution is updated automatically with time according to flow characteristics in the modeling process with higher resolution around the free surface and the solid boundary and lower resolution in less important area.The model has good performance in saving computer memory and CPU time and is validated by computational examples of small amplitude wave,second-order Stokes wave and cnoidal wave.Computational results also indicate that standing wave and wave overtopping are also reasonably simulated by the model.     

A Time-Domain Coupled Model for Nonlinear Wave Forces on A Fixed Box-Shaped Ship in A Harbor

A 2-D time-domain numerical coupled model is developed to obtain an efficient method for nonlinear wave forces on a fixed box-shaped ship in a harbor.The domain is divided into an inner domain and an outer domain.The inner domain is the area beneath the ship and the flow is described by the simplified Euler equations.The other area is the outer domain and the flow is defined by the higher-order Boussinesq equations in order to consider the nonlinearity of the wave motions.Along the interface boundaries between the inner domain and the outer domain,the volume flux is assumed to be continuous and the wave pressures are equal.Relevant physical experiment is conducted to validate the present model.It is shown that the numerical results agree with the experimental data.Compared with the coupled model with the flow in the inner domain governed by the Laplace equation,the present coupled model is more efficient and its solution procedure is more simple,which is particularly useful for the study on the effect of the nonlinear wave forces on a fixed box-shaped ship in a large harbor.