考虑边界波浪方向的缓坡方程自适应求解模型

缓坡方程是描述近岸波浪运动较好的数学模型之一。在发展的自适应有限元求解缓坡方程的基础上,采用迭代求解的方法,确定波浪相对于边界的入射方向,从而对边界条件进行改进,建立了求解缓坡方程的数值计算模型。典型算例表明,考虑波浪相对于边界的入射角度后,模型可以更好地模拟吸收波浪边界,同时对多向波对双突堤的绕射进行了模拟研究,与试验结果比较表明,所建立的数值计算模型能够适用于多向不规则波传播过程的模拟研究。     

A finite difference method for effective treatment of mild-slope wave equation subject to non-reflecting boundary conditions

A numerical model for coastal water wave motion that includes an effective method for treatment of non-reflecting boundaries is presented. The second-order one-way wave equation to approximate the non-reflecting boundary condition is found to be excellent and it ensures a very low level of reflection for waves approaching the boundary at a fairly wide range of the incidence angle. If the Newman approximation is adopted, the resulting boundary condition has a unique property to allow the free propagation of wave components along the boundary. The study is also based on a newly derived mild-slope wave equation system that can be easily made compatible to the one-way wave equation. The equation system is theoretically more accurate than the previous equations in terms of the mild-slope assumption. The finite difference method defined on a staggered grid is employed to solve the basic equations and to implement the non-reflecting boundary condition. For verification, the numerical model is then applied to three coastal water wave problems including the classical problem of plane wave diffraction by a vertical circular cylinder, the problem of combined wave diffraction and refraction over a submerged hump in the open sea, and the wave deformation around a detached breakwater. In all cases, the numerical results are demonstrated to agree very well with the relevant analytical solutions or with experimental data. It is thus concluded that the numerical model proposed in this study is effective and advantageous.     

An energy-controlling boundary condition for partial wave reflections in the mild slope equation

An energy-controlling technique to actively manage the reflective property of waves from solid boundary is presented. As linear waves propagate through an energy-controlling area, a reduction in wave heights occurs due to energy dissipation, which can be placed under direct control through the imaginary part of the wavenumber and phase velocity. Based on this relationship, the present study investigates a new method to control reflected waves with desired heights in the mild slope equation model. The method is validated through numerical tests for various reflection coefficients and the results confirm the promising use of energy-controlling boundary condition for partial wave reflections.     

修正型缓坡方程的有限元模型

与缓坡方程相比,修正型缓坡方程增加了地形曲率项和坡度平方项,从而提高了数值求解的复杂性。本文将计算域划分为内域和外域,内域为水深变化区域,使用修正型缓坡方程,其中的地形曲率项和坡度平方项可用有限单元各节点的水深信息和单元插值函数表示,外域为水深恒定区,速度势满足Helmholtz方程,通过内外域的边界匹配建立有限元方程,并用高斯消去法求解。进而分别模拟了波浪传过Homma岛和圆形浅滩的变形,其结果与相关的解析解和实验数据吻合良好,证明了本文有限元模型的正确性。同时,通过与实验数据的对比也明显看出,在地形坡度较陡的情况下,修正型缓坡方程较缓坡方程具有更高的计算精度。     

Assessment of open boundary conditions on the elliptic formulation of the mild-slope equation

The performance of open boundaries in a finite differences scheme of the elliptic mild-slope equation is assessed. The wave propagation results show that lowest order parabolic radiation boundary conditions, unlike sponge layers combined with first order radiation boundary conditions, are an efficient alternative to first order radiation boundary conditions in order to improve the accuracy of the numerical solution of the problem.     

Open boundary conditions for regional tidal simulations

The performance of regional tide model simulations is examined in relation to the choice of open boundary conditions. Three barotropic open boundary conditions, clamped elevation, clamped normal velocity, and Flather, give similar results when the prescribed values are exact; however, Flather is much less sensitive to errors in the prescribed values. Of particular concern, it was found that with a phase error between the two boundaries, both the clamped conditions resulted in magnitude errors in the unclamped variable (although the simulation remained stable).A modified flow relaxation scheme for the depth-varying prognostic variables is presented. This implementation allows the transmission of a range of vertical modes while retaining realistic topography at the boundary. It was found to be an excellent internal tide boundary condition in tests comparing simulations of different domain length encompassing a ridge and sloping bottom, and in a comparison to an analytical solution. Mass is conserved without any artificial volume constraint.     

Active open boundary forcing using dual relaxation time-scales in downscaled ocean models

Regional models actively forced with data from larger scale models at their open boundaries often contain motion at different time-scales (e.g. tidal and low frequency). These motions are not always individually well specified in the forcing data, and one may require a more active boundary forcing while the other exert less influence on the model interior. If a single relaxation time-scale is used to relax toward these data in the boundary equation, then this may be difficult. The method of fractional steps is used to introduce dual relaxation time-scales in an open boundary local flux adjustment scheme. This allows tidal and low frequency oscillations to be relaxed independently, resulting in a better overall solution than if a single relaxation parameter is optimized for tidal (short relaxation) or low frequency (long relaxation) boundary forcing. The dual method is compared to the single relaxation method for an idealized test case where a tidal signal is superimposed on a steady state low frequency solution, and a real application where the low frequency boundary forcing component is derived from a global circulation model for a region extending over the whole Great Barrier Reef, and a tidal signal subsequently superimposed.     

Linear and nonlinear irregular waves and forces in a numerical wave tank

A time-domain higher-order boundary element scheme was utilized to simulate the linear and nonlinear irregular waves and diffractions due to a structure. Upon the second-order irregular waves with four Airy wave components being fed through the inflow boundary, the fully nonlinear boundary problem was solved in a time-marching scheme. The open boundary was modeled by combining an absorbing beach and the stretching technique. The proposed numerical scheme was verified by simulating the linear regular and irregular waves. The scheme was further applied to compute the linear and nonlinear irregular wave diffraction forces acting on a vertical truncated circular cylinder. The nonlinear results were also verified by checking the accuracy of the nonlinear simulation.     

基于抛物型缓坡方程模拟近岸植被区波浪传播

植被对波浪传播运动有重要影响。考虑近岸波浪在植被区传播中的折射、绕射、破碎及植被引起的波能耗损效应,基于抛物型缓坡方程建立了模拟近岸植被区波浪传播的数学模型,对模型进行了数值模拟验证,采用数值模拟试验分析了植被对波浪传播的影响。数值模拟结果表明,波浪在近岸植被区传播时,随着植被密度和植被高度的增加,波浪传播中的波高衰减增大,波能耗损增加;不同周期波浪在植被区传播中的波高衰减过程也明显不同。     

Simulation of small-amplitude frequency-dispersive transient waves by means of the mild-slope equation

A numerical model based on the mild-slope equation is applied to reproduce the propagation of small-amplitude transient waves. The model makes use of the Fourier Transform to convert the time-dependent hyperbolic equation into a set of elliptic equations in the frequency domain. The results of two available experimental studies on tsunamis generated by landslides are used to validate the model, which appears to be able of carefully reproducing the effects of the frequency dispersion. An example application of tsunamis propagating around the Stromboli island is also presented to show the applicability of the present approach to real life scenarios. It is finally discussed how this model could be applied as support to a tsunami early warning system.     

The influence of seaward boundary conditions on swash zone hydrodynamics

The influence of the seaward boundary condition on the internal swash hydrodynamics is investigated. New numerical solutions of the characteristics form of the nonlinear shallow-water equations are presented and applied to describe the swash hydrodynamics forced by breaking wave run-up on a plane beach. The solutions depend on the specification of characteristic variables on the seaward boundary of the swash zone, equivalent to prescribing the flow depth or the flow velocity. It is shown that the analytical solution of Shen and Meyer [Shen, M.C., Meyer, R.E., 1963. Climb of a bore on a beach. Part 3. Runup. J. Fluid Mech. 16, 113–125] is a special case of the many possible solutions that can describe the swash flow, but one that does not appear appropriate for practical application for real waves. The physical significance of the boundary conditions is shown by writing the volume and momentum fluxes in terms of the characteristic variables. Results are presented that illustrate the dependence of internal flow depth and velocity on the boundary condition. This implies that the internal swash hydrodynamics depend on the shape and wavelength of the incident bore, which differs from the hydrodynamic similarity inherent in the analytical solution. A solution appropriate for long bores is compared to laboratory data to illustrate the difference from the analytical solution. The results are important in terms of determining overwash flows, flow forces and sediment dynamics in the run-up zone.     

A method to determine the incident wave boundary conditions and its application

When studying the harbor water tranquility, cases are often confronted as that the verification point is not located on the generation line or that the angle between the generation line and the isobath is so large that the differences of the wave climates along the generation line can not be ignored. For these cases, the incident boundary conditions are difficult to evaluate. In order to solve this problem, a combined wave model is developed in the present paper based on the Boussinesq equation and the wave action balance equation. Instead of the one-line wave generation method, a multi-line generation method is proposed for the combined model. Application of this method is given to a case that the harbor is designed with two entrances and the angle between the generation line and the isobath is large and the results are shown reasonable. We suggest that the wave generation method on multi-lines might also be introduced to the wave physical model as the replacement for the one-line generation method.     

高阶边界元方法求解规则波在三维局部渗透海床上的传播

基于线性势流理论,利用高阶边界元法研究了规则波在三维局部渗透海床上的传播。根据Darcy渗透定律推导出渗透海床的控制方程,利用渗透海床顶部和海底处法向速度和压强连续条件得到渗透海床顶部满足的边界条件。根据绕射理论,利用满足自由水面条件的格林函数建立了求解渗透海床绕射势的边界积分方程,采用高阶边界元方法求解边界积分方程进而得到自由水面的绕射势和波浪在局部渗透海床上传播过程中幅值的变化情况。通过与已发表的波浪对圆柱形暗礁的时域全绕射结果对比,证明了本文建立的频域方法计算波幅的正确性和有效性。利用这一模型研究了三维矩形渗透海床区域上波浪的传播特性,并分析了入射波波长、海床渗透特性系数等参数对波浪传播的影响。     

利用伴随法优化非线性潮汐模型的开边界条件Ⅰ．伴随方程的建立及“孪生"数值试验

本研究采用基于最优控制理论的伴随法,把观测资料同化到陆架海域潮汐数值模型中去,优化开边界条件,提高数值预报的精度．潮汐模型的控制方程为考虑平流项、非线性底摩擦和侧向涡动粘性项的非线性浅水方程组;采用Lagrange乘子法建立了伴随模型．研究分两部分：第一部分即本文,建立非线性浅水方程模型的伴随方程、给出目标函数的梯度,并实现“孪生”数值试验;第二部分另文给出．     

Improvement on open boundaries on a time dependent numerical model of wave propagation in the nearshore region

An improvement on the simulation of outgoing waves on a time dependent numerical model for water wave propagation in the nearshore region is presented. The governing equations consist of a system of first order partial differential equations (PDEs), the equation of continuity and the equation of motion. A comparative study of first order radiation boundary conditions (BCs) and first order radiation BCs combined with sponge layers is presented for cases where outgoing waves leave the numerical domain of calculation through the open boundary. A reduction of spurious reflections from the numerical open boundaries can be obtained with an irrelevant increase in terms of computational cost.     

The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies

A coupled finite element (FEM) and boundary element (BEM) method is developed to analyse the nonlinear interaction between bodies and water waves. The former is used away from the body while the latter is used in a region near body. The combination is based on consideration of the efficiency of FEM and BEM in computation and mesh generation, respectively. Results for wave/body interactions are obtained by using auxiliary functions to decouple the mutual dependence of the body acceleration and the wave force.     

An analytic solution to the wave bottom boundary layer governing equation under arbitrary wave forcing

Existing models of the wave bottom boundary layer have focused on the vertical and temporal dynamics associated with monochromatic forcing. While these models have made significant advances, they do not address the more complicated dynamics of random wave forcing, commonly found in natural environments such as the surf zone. In the closed form solution presented here, the eddy viscosity is assumed to vary temporally with the bed shear velocity and linearly with depth, however, the solution technique is valid for any eddy viscosity which is separable in time and space. A transformation of the cross-shore velocity to a distorted spatial domain leads to time-independent boundary conditions, allowing for the derivation of an analytic expression for the temporal and vertical structure of the cross-shore velocity under an arbitrary wave field. The model is compared with two independent laboratory observations. Model calculations of the bed shear velocity are in good agreement with laboratory measurements made by Jonsson and Carlsen (1976, J. Hydraul. Res., 14, 45–60). A variety of monochromatic, skewed, and asymmetric wave forcing conditions, characteristic of those found in the surf zone, are used to evaluate the relative effects on the bed shear. Because the temporal variation of the eddy viscosity is assumed proportional to the bottom shear, a weakly nonlinear interaction is created, and a fraction of the input monochromatic wave energy is transferred to the odd harmonics. For a monochromatic input wave, the ratio of the third harmonic of velocity at the bed to the first is <10%. However, for a skewed and asymmetric input wave, this ratio can be as large as 30% and is shown to increase with increasing root-mean-square input wave acceleration. The work done by the fluid on the bed is shown to be a maximum under purely skewed waves and is directed onshore. Under purely asymmetric waves, the work done is significantly smaller and directed offshore.     

A novel high resolution model without open boundary conditions applied to the China Seas: first investigation on tides

We developed a Global Ocean Circulation and Tide Model (GOCTM) with coarse grids in the open deep ocean degrading ‘smoothly’ into the highly resolved China Seas (CS) of refined grids to study the tides and circulation there.GOCTM is based on the framework of the Finite Volume approach for better mass conservation through improved transports across the discrete individual control volume.It also takes a full advantage of the geometric flexibility of unstructured mesh using a realistic global topography including the Arctic Ocean.The CS are given a special focus by refining the unstructured grids,but they are embedded into global domain naturally.Furthermore,GOCTM not only successfully avoids the treatment of the open boundaries,but also optimizes the trade-off between computational cost and model accuracy.Meanwhile,GOCTM is driven by the astronomical tide-generating potential and the secondary tide-generating potential directly,together with the wind stress and heat flux.GOCTM succeeds in reproducing the global eight principal tidal harmonic constants.Particularly,the simulated tidal results in the CS are improved compared to some other regional models with the discrepancy of 3.9 cm for M 2 tide.This idea of GOCTM can also be referred for other regional ocean study.     

消除"不规则频率"的非连续高阶元方法

针对使用边界元法计算波浪与结构物相互作用时所出现的“不规则频率”现象，采用连续高阶元和部分非连续高阶元对通过修改积分区域所获得的边界积分方程进行离散，有效地消除了“不规则频率”现象的发生。波浪作用下的截断圆柱所受到的水平波浪力和垂向波浪力的数值计算结果验证了该方法的有效性，同时考虑了非连续单元配置点的选择及单元划分数目对消除效果的影响。