Interaction of free-surface waves with a floating dock

A new method to describe the interaction of waves with a rigid or flexible dock, with zero draft, is derived. By means of Green's theorem an integral equation along the platform for either the velocity potential or the deflection is obtained. In the two-dimensional case this equation is solved by means of a superposition of exponential functions. With a specific choice of the Green function the integration with respect to the space coordinate can be carried out analytically. The integration left is the integration in the k-plane that occurs in the chosen Green function. Subsequently the contour of this integral is modified in the complex plane. This results at first in a dispersion relation for the phase functions in the expansion. Then the set of algebraic equations for the amplitude coefficients follows from the same singularity analysis in the complex plane. These equations are very simple and easy to solve. In contrast to the classical approach of eigen-mode expansions, there is no need to split the problem in a symmetric and antisymmetric one. An other advantage is that the transmission and reflection coefficients are determined seperately by means of Green's theorem, applied at the free surface in the far field. The method is first explained for the semi-infinite rigid dock, followed by the rigid strip, the moving strip and the flexible moving platform. In the appendix it is explained how to derive a set of algebraic equations in the case when the incident wave is not perpendicular to the strip.     

On the use of boundary-integral equation methods for unsteady free-surface flows

The Mixed Eulerian-Lagrangian Methods (MEL) forfree-surface potential flows solved by boundary-integral equations (BIEs) is considered, and the diffusion and dispersion errors are studied in the discrete linearized problem. The diffusion error is the base for the stability analysis of the scheme; both the errors give indications on the accuracy of the numerical solution. The study is divided into two steps: comparison of the discrete dispersion relation with the analytical solution and coupling with different time-integration schemes. In particular, a stability analysis of the Runge-Kutta and Taylor-expansion schemes, previously not given in the literature, is addressed. It is shown that MEL methods based on first- and second-order explicit Runge-Kutta and Taylor-expansion schemes are unstable, regardless of the technique adopted to discretize the BIEs. Higher-order Runge-Kutta and Taylor-expansion schemes lead to conditionally stable methods. Known results for explicit, implicit and explicit-implicit Euler schemes are recovered by the present analysis. The theoretical predictions of the errors are confirmed for two different boundary-element techniques: a high-order panel method based on B-Splines to solve for the velocity potential and a spectrally-accurate method based on the Euler-McLaurin summation formula to solve directly for the velocity field.     

Generation of water waves and bores by impulsive bottom flux

The inviscid free-surface flow due to an impulsive bottom flux on constant depth is investigated analytically and numerically. The following classes of two-dimensional flow are considered: an upwelling flow which is uniform over a half-plane, a line source/sink, and a dipole aligned along the bottom. The bottom flux is turned on impulsively and may decay with time. The fully nonlinear problem is solved numerically. A small-time asymptotic expansion to third order is found for the nonlinear problem. An asymptotic large-time solution is found for the linearized problem. A steady source will generate a pair of symmetric bores, and their breaking is investigated. A steady sink generates a depression wave if it is weak, and dip instability if it is strong. Wave breaking will occur for intermediate sink strengths. A decaying source emits solitary waves.     

Free-surface wave interaction with a thick flexible dock or very large floating platform

In this paper the recently developed semi-analytic method to solve the free-surface wave interaction with a thin elastic plate is extended to the case of a plate of finite thickness. The method used is based on the reformulation of the differential–integral equation for this problem. The thickness of the plate is chosen such that the elastic behavior of the plate can be described by means of thin-plate theory, while the water pressure at the plate is applied at finite depth. The water depth is finite.     

Nonlinear capillary free-surface flows

Capillary free-surface flows are considered. The fluid is taken to be inviscid and incompressible and the flow to be irrotational. Particular attention is devoted to two-dimensional flows for which the free surfaces intersect rigid walls. These include cavitating flows and local flows at the front of a small object (probe or insect) moving at the surface of a fluid. A general study of the effect of surface tension on the possible singularities which can occur at the separation points is presented. The results confirm and generalise previous findings on the subject.     

Breaking and merging of liquid sheets and filaments

The surface of a sheet of liquid which contracts due to surface tension, breaks, and then pulls apart into two pieces, is calculated. Before breaking, the flow is the self-similar one found by Keller, Milewski and Vanden-Broeck. After breaking, it is the self-similar flow found by Keller and Miksis. A general numerical scheme, which includes the previous ones, is presented and new numerical results are discussed. There is an analogous flow of an axially symmetric liquid filament, but it is not calculated.     

Hydroelastic behaviour of compound floating plate in waves

The paper deals with the plane problem of the hydroelastic behaviour of floating plates under the influence of periodic surface water waves. Analysis of this problem is based on hydroelasticity, in which the coupled hydrodynamics and structural dynamics problems are solved simultaneously. The plate is modeled by an Euler beam. The method of numerical solution of the floating-beam problem is based on expansions of the hydrodynamic pressure and the beam deflection with respect to different basic functions. This makes it possible to simplify the treatment of the hydrodynamic part of the problem and at the same time to satisfy accurately the beam boundary conditions. Two approaches aimed to reduce the beam vibrations are described. In the first approach, an auxiliary floating plate is added to the main structure. The size of the auxiliary plate and its elastic characteristics can be chosen in such a way that deflections of the main structure for a given frequency of incident wave are reduced. Within the second approach the floating beam is connected to the sea bottom with a spring, the rigidity of which can be selected in such a way that deflections in the main part of the floating beam are very small. The effect of the vibration reduction is quite pronounced and can be utilized at the design stage.     

Starting flow generated by the impulsive start of a floating wedge

The initial stage of the plane unsteady flow caused by the impulsive vertical motion of a wedge initially floating on an otherwise flat free surface is investigated with the help of a combination of numerical and asymptotic methods. The liquid is assumed ideal and incompressible and its flow potential. Compressible effects give a negligible contribution to the flow close to the entering body at the stage considered in the present paper. The vertical velocity of the body is constant after the impulsive start. The flow domain is divided into an outer region, where the first-order solution is given by the pressure-impulse theory, and inner regions close to the intersection points between the free surface and the moving body. The relative displacement of the body plays the role of a small parameter. The inner solution is matched with the outer one. The outer solution is given in quadratures but the inner solution, which is shown to be nonlinear and self-similar, can be found only numerically. With the aim of deriving the inner solution, the inner region is divided into three parts. In the far-field zone the solution is given in terms of its asymptotic behavior while, in the jet region, attached to the wedge, the flow is described by a second-order shallow-water approximation. In the intermediate region a boundary-element method is used, which is suitably coupled with the solutions in both the jet and the far-field regions through an iterative pseudo-time stepping procedure. The procedure is dependent on the deadrise angle of the wedge. If the angle is equal or smaller than π/4, eigensolutions appear in the far-field asymptotics and their amplitudes are recovered together with the solution. The approach is applied to different values of the wedge deadrise angle. The obtained results can be used to improve the prediction of the hydrodynamic loads acting on floating bodies, the velocity of which changes rapidly.     

The effect of grease on the activated sludge process in wastewater treatment

Abstract

The purpose of this paper is to introduce the major potentials and perspectives of applications of finite element analysis in solving the problems of shallow water wave equations. One‐dimensional and two‐dimensional shallow water wave equations will both be incorporated into the modeling procedures. For one‐dimensional flows, the models will cover the typical single channels, confluence channels system, division channels system, and natural river systems. As far as two‐dimensional flows are concerned, the overland flows are investigated. The simulation results are compared with the data obtained by physical modeling and field observation and with the results of other existing literature. The models were found to be very feasible in modeling the complex flow fields of shallow water wave equation problems.     

漂浮式光伏发电装置在海浪影响下的光照性能研究

针对海洋漂浮式太阳能发电装置,为分析海浪运动对其光照性能的影响,对比陆地环境下相同面积光伏电池的光照性能,分析两者之间的差距.首先总结了倾斜表面太阳辐射能的计算方法,分析了在海洋环境下应用这一模型的差异性,再将海浪简化,分析浮体在海浪中的运动特性及角度摆动规律,同时运用AQWA仿真软件,对浮体运动进行仿真,以验证浮体摆动角度计算的准确性,最后将静态的太阳辐射能计算模型与动态的浮体角度摆动相结合,得出海浪影响下光伏电池表面所接收到的太阳辐射能.将其与相同条件下以最佳接收角度固定安装的光伏电池所接收的太阳辐射能相比较,得出在海浪影响下浮体及光伏电池所接收的辐射能能够达到陆地最佳接收情况下的87.5％.     

柔性ITO衬底上铜酞菁薄膜有序生长的X射线衍射研究

采用石英晶体微天平实时监测薄膜生长速率,通过控制衬底温度与薄膜生长速率,在柔性ITO导电衬底上真空蒸发沉积了铜酞菁薄膜.X射线衍射分析表明,适当提高衬底温度与薄膜生长速率,可促进薄膜的有序生长.当衬底温度为90℃,生长速率为10nm/min时,薄膜的有序度最高,薄膜晶型呈(相和(200)晶面.     

Inverse-heat-conduction problems with boundary elements: analysis of a corner effect

A transient two-dimensional inverse-heat-conduction problem is investigated. It consists in the determination of both temperature and heat-flux density in the vicinity of an angle (≤ 180°) when some internal temperatures are known. Such a problem is solved by using a boundary-element approach with a time- and space-dependent fundamental solution. It uses a time-marching scheme that involves future time steps and a regularization procedure. An exhaustive study of sensitivity to the leading parameters of the problem is produced, and it is especially pointed out that the accuracy of the resolution is strongly affected as the corner angle decreases. In order to overcome this difficulty, a localized regularization procedure is suggested.     

Subsonic slip waves along the interface between two piezoelectric solids in sliding contact with local separation

The Stroh formalism of piezoelectric crystals and singular integral equation technique are applied to study the propagation of possible slip waves in presence of local separation at the interface between two frictionless contact piezoelectric solids, which are pressed together by uniaxial pressure and laid in the electric field. The problem is cast into a set of singular integral equations of which the closed solutions are obtained. Discussion on the existence of such slip waves is presented. The results show that such slip waves, which have square-root singularities at both ends of the local separation zones, can propagate in some special material combinations. And the existence of such slip waves is related with the applied mechanical and electric fields.     

Applicability of the method of fundamental solutions to interaction of fully nonlinear water waves with a semi-infinite floating ice plate

In this research effort, a meshless numerical model was developed to study the hydroelastic interaction of an incident wave with a semi-infinite horizontal floating plate. It is assumed that the fluid is homogenous, inviscid and incompressible. Fundamental solution of the governing Laplace equation is considered to be radial basis functions. In this method, only a few boundary points are located on the boundary. Moreover, there are a few source points that are located outside the computational domain. Two additional source points are introduced to deal with the plate's edge conditions. The problem is solved using collocations at only a few boundary points. When density and thickness of the plate are reduced to zero, good agreements with other numerical works are observed.     

Modelling of progressive short waves using wave envelopes

We consider progressive waves such that the time independent potential satisfies the Helmholtz equation, for example, the travelling wave diffracted from a body. In order to model the wave potential using finite elements it is usual to discretize the domain such that there are about ten nodal points per wavelength. However, such a procedure is computationally expensive and impractical if the waves are short. The goal is to be able to model accurately with few elements problems such as sonar and radar. Therefore we seek a new method in which the discretization of the domain is more economical. To do so, we express the complex potential ϕ in terms of the real wave envelope A and the real phase p such that ϕ=Ae^ip, and expect that in most regions the functions A and p vary much more gradually over the domain than does the oscillatory potential ϕ. Therefore instead of modelling the potential we model the wave envelope and the phase. The usual approach then uses the well known geometrical optics approximation (see p. 109 of Reference 1) : if the wave number k is large then the potential can be expanded in decreasing powers of k. The first two terms give the eikonal equation for the phase and the transport equation for the wave envelope respectively (see p. 149 of Reference 2). However, using the geometrical optics approximation (or ray theory) gives no diffraction effects. This approach shall therefore not be considered. (We note though that Keller's theory of geometrical diffraction, an extension to geometrical optics, does allow for diffraction effects and this may be considered at a later date.) We shall consider a new method which shall be described in the present paper and apply it to two-dimensional problems, although the method is equally valid for arbitary three-dimensional problems. (The method has already been validated for the case of one-dimensional problems.) An iterative procedure is described whereby an estimate of the phase is first given and from the resulting finite element calculation for the wave envelope a better estimate for the phase is obtained. The iterated values for the phase and wave envelope converge to the expected values for the test progressive wave examples considered. Even if a very poor estimate for the phase is first given the iterated values converge to the exact values but very slowly. © 1997 John Wiley & Sons, Ltd.     

A class of model equations for bi-directional propagation of capillary-gravity waves

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter α=a/h₀ and wavelength parameter β=(h₀/l)², where a and l are the actual amplitude and wavelength of the surface wave, and h₀ is the height of the undisturbed water surface from the flat bottom topography. These equations are also characterized by the surface tension parameter, namely the Bond number τ=Γ/ρgh₀², where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity.The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) β?(1/3−τ)?1/3 and (ii) (1/3−τ)=O(β). The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for τ=1/3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation.     

Investigation of the scattering of harmonic elastic antiplane shear waves by a finite crack using the non-local theory

In this paper, the scattering of harmonic antiplane shear waves by a finite crack is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. Contrary to the classical elasticity solution, it is found that no stress singularity is presented at the crack tip. The non- local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length. This revised version was published online in July 2006 with corrections to the Cover Date.     

作大范围运动的柔性梁的动力学分析

对附着在空间运动体上的柔性悬臂梁的动力学进行了研究，利用微元法建立了中心刚体作任意三维大位移运动时柔性悬臂梁作横向和纵向振动的动力学方程，此动力学方程计及了动力刚化效应。在对柔性梁离散求解时考虑了横向弯曲对纵向变形的影响，最后通过几个例子分析了运动基上柔性梁的动力学行为。     

Control of a 2-DOF manipulator with a flexible forearm

In this paper we present experimental results on the position and vibration control of the end-effector of a 2-DOF parallelogram manipulator with a flexible forearm. A dynamic model of the manipulator is first obtained. Control strategies are implemented to control the manipulator. The first control strategy uses the computed torque method based on a reduced-order dynamic model of the manipulator which is obtained by assuming that all the links are rigid. This method is referred to as the reduced-order computed torque (ROCT) method. Experimental results demonstrate that such a strategy is not good for vibration control of the end-effector of the manipulator. The second control strategy is a state feedback control law designed based on a local linearization of the nonlinear dynamic model. Experimental results show that this control strategy achieves good vibration control of the end-effector of the manipulator. However, since the strategy is based on local linearization, it is valid only in a neighbourhood of the operating point. A hybrid controller that uses the ROCT method for the initial large movement of the manipulator is then implemented. Based on a switching rule, the controller is switched to the state feedback controller based on the linearized model when the manipulator is sufficiently close to the equilibrium state. Experimental results are reported and the successful performance of the controller in dampening out end-point vibrations is demonstrated.