On the oblique water-entry problem of a rigid sphere

The case of oblique water-entry of a rigid sphere into an ideal incompressible fluid is studied analytically in order to determine the hydrodynamical loads acting on the body. We consider the motion imparted to the fluid by an impulsively-started partially-submerged sphere under the large-impact approximation, in which the free surface is assumed flat and equipotential. Asymptotic small-time expressions are derived for both the vertical and horizontal time-dependent added masses and analytical expressions for the hydrodynamic forces are obtained by differentiating these added masses with respect to the instantaneous submergence depth. The resulting expressions are also compared with corresponding numerical solutions and with a known solution for a two-dimensional profile.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Hydrodynamics of deformable contiguous spherical shapes in an incompressible inviscid fluid

Summary An exact solution to the two-body interaction problem is presented for the case of spherical shapes moving in an incompressible and inviscid fluid. The spheres are assumed to translate in an arbitrary manner and to undergo radial deformation (or pulsation). The problem is formulated in terms of spherical harmonics and the force experienced by the spheres is obtained by employing the Lagally theorem. The expressions for the force are given as an infinite sum of coefficients which are found by solving an infinite set of linear equations. Three main geometries are considered, namely, two spheres exterior to each other, one sphere in the interior of the other and sphere in a rectangular channel. Numerical values for the added-mass coefficients as well as for the hydrodynamic forces are found for the case of rigid sphere moving toward or parallel to a rigid wall or a free surface, and a pulsating sphere in the proximity of these boundaries. Also given are numerical values for the transverse and the longitudinal addedmass coefficients for a sphere moving in a rectangular channel for different channel-blockage ratios.     

Wave radiation and diffraction by a two-dimensional floating rectangular body with an opening in its bottom

The radiation and diffraction problem of a two-dimensional rectangular body with an opening in its bottom floating on a layer of water of finite depth is analysed based on the linearized velocity potential theory through an analytical solution procedure. The expressions for the potentials are obtained by the method of separation of variables, in which the unknown coefficients are determined by the boundary condition and matching requirement on the interface. The hydrodynamic coefficients and wave excitation forces are obtained and verified using the near-field and far-field methods and the symmetry properties of coupled hydrodynamic coefficients. The effect of the opening on the wave excitation force and hydrodynamic coefficients is investigated. Piston resonant behaviour and sloshing resonant behaviour are also investigated and their effect on the wave excitation force and hydrodynamic coefficients is discussed.     

Water-wave scattering by two symmetric circular-arc-shaped thin plates

The problem of surface water-wave scattering by two symmetric circular-arc-shaped thin plates submerged in deep water is investigated in this paper assuming linear theory. The problem is formulated in terms of hypersingular integral equations which are solved approximately using finite series involving Chebyshev polynomials of the second kind. The coefficients of the finite series are obtained numerically by a collocation method. Very accurate numerical estimates for the reflection and the transmission coefficients are then obtained. The numerical results are depicted graphically against the wave number for different arc lengths of the plates, the depth of their submergence and the separation length. Known results for a circular cylinder and horizontal straight plate are recovered.     

Irrotational axisymmetric flow about a prolate spheroid in cylindrical duct

Summary A solution to the problem of potential flow about a prolate spheroid placed axially symmetric in a circular duct has been derived. The solution is in the form of a distribution of vortex rings over the surface of the spheroid. The vortex strength is expressed in terms of an infinite series of Legendre polynomials and the analysis yields an infinite set of equations for determining the coefficients of this series. An expression for the velocity distribution on the surface of the spheroid as well as the longitudinal added mass coefficients of the spheroid are derived in terms of the coefficients of the Neumann series expansion of the vortex sheet strength. Numerical results are presented for various spheroids and different blockages. Also given is a comparison between the present method and few available approximate methods.     

Exact and numerical elastodynamic solutions for spherically symmetric problems of functionally graded thick-walled spheres subjected to pressure shocks

In the present paper, analytical and numerical elastodynamic solutions are developed for spherically symmetric problems of functionally graded thick-walled spheres subjected to arbitrary dynamic and shock loads. Both transient dynamic response and elastic wave propagation characteristics are studied in the mentioned nonhomogeneous structures. Variations of the material properties across the thickness are described according to both polynomial and power law functions. The numerical consistent transfinite element formulation is presented for both functions whereas the exact solution is presented for the power law function. The functionally graded material sphere is not divided into isotropic sub-spheres. An approach associated with dividing the dynamic radial displacement expression into quasi-static and dynamic parts and expansion of the transient wave functions in terms of a series of eigenfunctions is employed to propose the exact solution. Results are obtained for various exponents of the functions of the material properties distributions, various radius ratios, and variety of dynamic and shock loads.     

Effect of a submerged vertical barrier on flexural gravity waves

An explicit solution is provided for the scattering of flexural gravity waves by a rigid vertical barrier submerged in an infinite depth of water. By applying recently developed mode-coupling relation for eigenfunctions, the mixed boundary value problem has been converted to solve dual integral equations with kernel consisting of trigonometric functions. And then complete analytical solutions are derived with an aid of singular integral equations whose solutions are bounded at the end points. The important hydrodynamical scattering quantities such as reflection and transmission coefficients associated with the flexural gravity wave scattering have been obtained analytically in terms of modified Bessel functions and Struve functions. It is observed that these quantities are sensitive to both combined as well as individual effect of plate thickness and barrier depth of submergence. Numerical results are computed and explained graphically for different parameters such as time period and non-dimensional wave length. Further, the effect of compressive force and plate thickness on the flexural gravity waves against a submerged vertical barrier is studied.     

On the computation of the nijboer-zernike aberration integrals at arbitrary defocus

Abstract

We present a new computation scheme for the integral expressions describing the contributions of single aberrations to the diffraction integral in the context of an extended Nijboer-Zernike approach. Such a scheme, in the form of a power series involving the defocus parameter with coefficients given explicitly in terms of Bessel functions and binomial coefficients, was presented recently by the authors with satisfactory results for small-to-medium-large defocus values. The new scheme amounts to systemizing the procedure proposed by Nijboer in which the appropriate linearization of products of Zernike polynomials is achieved by using certain results of the modern theory of orthogonal polynomials. It can be used to compute point-spread functions of general optical systems in the presence of arbitrary lens transmission and lens aberration functions and the scheme provides accurate data for any, small or large, defocus value and at any spatial point in one and the same format. The cases with high numerical aperture, requiring a vectorial approach, are equally well handled. The resulting infinite series expressions for these point-spread functions, involving products of Bessel functions, can be shown to be practically immune to loss of digits. In this respect, because of its virtually unlimited defocus range, the scheme is particularly valuable in replacing numerical Fourier transform methods when the defocused pupil functions require intolerably high sampling densities.     

Scattering of a surface wave by a submerged sphere

Summary The problem of the scattering of a surface wave in a nonviscous, incompressible fluid of infinite depth by a fully submerged, rigid, stationary sphere has been reduced to the solution of an infinite set of linear algebraic equations for the expansion coefficients in spherical harmonics of the velocity potential. These equations are easily solved numerically, so long as the sphere is not too close to the surface. The approach has been to formulate the problem as an integral equation, expand the Green's function, the velocity potential of the incident wave, and the total velocity potential in spherical harmonics, impose the boundary condition at the surface of the sphere, and carry out the integrations. The scattering cross section has been evaluated numerically and is shown to peak for values of the product of radius and wave number somewhat less than unity. Also, the Born approximation to the cross section is obtained in closed form.Supported by the Department of the Navy, Naval Sea Systems Command under Contract No. N00017-72-C-4401.     

A higher-order theory for functionally graded beams based on Legendre's polynomial expansion

In this article a higher-order theory for functionally graded beams based on the expansion of the two-dimensional (2D) equations of elasticity for functionally graded materials into Fourier series in terms of Legendre's polynomials is presented. Starting from the 2D equations of elasticity, the stress and strain tensors, displacement, traction, and body force vectors are expanded into Fourier series in terms of Legendre's polynomials in the thickness coordinate. In the same way, the material parameters that describe the functionally graded material properties are also expanded into Fourier series. All equations of the linear elasticity including Hooke's law are transformed into the corresponding equations for the Fourier series expansion coefficients. Then a system of differential equations in terms of the displacements and the boundary conditions for the Fourier series expansion coefficients are obtained. In particular, the first- and second-order approximations of the exact infinite dimensional beam theory are considered in more detail. The obtained boundary-value problems are solved by the finite element method with MATHEMATICA, MATLAB, and COMSOL multiphysics software. Numerical results are presented and discussed.     

沿水平向运动的水中截断圆柱体辐射波浪问题的解析解答

根据线性势波理论,分析了水中截断圆柱体作水平简谐运动时结构周围的辐射波浪。利用分离变量法,分别得到了含有未知常数的三个流体子域速度势的简谐表达形式,并采用一个较为简单的匹配方法使其在流体子域的共同边界上满足压力和法向速度的连续条件。于是求解得到了速度势,进而可得到由等效附加质量和附加阻尼表示的柱体侧面上的动水力。不仅能考虑自由表面波对动水压力的影响,也适用于位于任意水深处的截断圆柱体。实例分析表明,该方法具有较高的计算精度;同时对于截断圆柱体,采用Morison方程中的动水附加惯性力项会高估柱体侧面上的动水力。     

Coefficients of the crack tip asymptotic field for wedge splitting specimens

The stress intensity factor (SIF) and the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are directly computed using a hybrid crack element. Accurate analytical expressions for the first five terms are obtained by fitting the computed data. Numerical results show that the coefficients of terms higher than three are negligibly small, this may explain that the wedge splitting specimen is more stable than other geometries. The first five terms are not sensitive to support conditions. However, for short cracks coefficients of terms, except the SIF, are quite sensitive to the loading arrangement even when the loads are statically equivalent.     

Application of response surfaces for reliability analysis of marine structures

Marine structures subjected to multiple environmental loads (i.e. waves, current, wind) are considered. These loads are characterized by a set of corresponding parameters. The structural fatigue damage and long-term response are expressed in terms of these environmental parameters based on application of polynomial response surfaces. For both types of analysis, an integration across the range of variation for all the environmental parameters is required. The location of the intervals which give rise to the dominant contribution for these integrals depends on the relative magnitude of the coefficients defining the polynomials. The required degree of numerical subdivision in order to obtain accurate results is also of interest. These issues are studied on a non-dimensional form. The loss of accuracy which results when applying response surfaces of too low order is also investigated. Response surfaces with cut-off limits at specific lower-bound values for the environmental parameters are further investigated.

Having obtained general expressions on non-dimensional form, examples which correspond to specific response quantities for marine structures are considered. Typical values for the polynomial coefficients, and for the statistical distributions representing the environmental parameters, are applied. Convergence studies are subsequently performed for the particular example response quantities in order to make comparison with the general formulation. For the extreme response, the application of ‘extreme contours’ obtained from the statistical distributions of the environmental parameters is explored.     

Wave-interference effects on a truncated cylinder in a channel

The effects of channel walls on the hydrodynamic properties of a floating vertical cylinder are examined. An interior eigensolution under the cylinder is matched with an exterior eigensolution in a manner similar to Yeung [1]. Wave effects due to an image cylinder can be conveniently expressed in terms of the coordinates of the central cylinder by the use of Graff's theorem. The infinite array results in a slowly convergent series which has to be summed with caution. Results for the heave added mass and damping of a cylinder for several geometric configurations are obtained. Also presented in the paper are results for the diffraction of incident waves about the same cylinder. The channel walls exert an important influence on the radiation and diffraction properties, the latter to a lesser extent. Such influence is characterized by the presence of spikes at wave frequencies corresponding to the occurrence of symmetric transverse resonant modes in the channel. An analytical solution of a three-dimensional flapper wavemaker given in the Appendix further confirms such characteristics. In the high-frequency range, the radiation properties approach those of a single cylinder. In the low-frequency limit, they exhibit a behavior similar to that of a two-dimensional horizontal cylinder heaving in water of finite depth.     

柱形地形上单浮体的辐射和散射问题

在有限水深、同轴但半径大于或等于浮体半径的圆柱体障碍物地形条件下，基于特征函数展开法，推导了垂直放置的圆柱形浮体由于波的辐射和散射作用所表现的动力学和运动学特征表达式，涉及浮体做垂荡、横荡和横摇运动所产生的辐射势，以及在入射波的作用下，由于浮体固定不动而产生的散射势，并推导了激励力、附加质量和阻尼系数表达式。采用与同轴、同半径圆柱体障碍物地形上单浮体水动力学特性相比的方式和激励力计算两种方法验证了推导的表达式，最后分析了障碍物几何尺寸对浮体水动力学特性的特有影响。     

Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid

The problem of a magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid is considered for the analytical solution using homotopy analysis method (HAM). The non-linear partial differential equations are transformed to an ordinary differential equation first taking boundary layer approximations into account and then using the similarity transformations. The analytical solution is presented in the form of an infinite series. The recurrence formulae for finding the coefficients are presented and the convergence is established. The effects of the Deborah number and MHD parameter is discussed on the velocity profiles and the skin friction coefficients. It is found that the results are in excellent agreement with the existing results in the literature for the case of hydrodynamic flow.     

Hydroelastic Behavior of a Floating Ring-Shaped Plate

The interaction between incident surface water waves and floating elastic plate is studied. This paper considers the diffraction of plane incident waves on a floating flexible ring-shaped plate and its response to the incident waves. An analytic and numerical study of the hydroelastic behavior of the plate is presented. An integro-differential equation is derived for the problem and an algorithm of its numerical solution is proposed. The representation of the solution as a series of Hankel functions is the key ingredient of the approach. The problem is first formulated. The main integro-differential equation is derived on the basis of the Laplace equation and thin-plate theory. The free-surface elevation, plate deflection and Green’s function are expressed in polar coordinates as superpositions of Hankel and Bessel functions, respectively. These expressions are used in a further analysis of the integro-differential equation. The problem is solved for two cases of water depth infinite and finite. For the coefficients in the case of infinite depth a set of algebraic equations is obtained, yielding an approximate solution. Then a solution is obtained for the general and most interesting case of finite water depth analogously in the seventh section. The exact solution might be approximated by taking into account a finite number of the roots of the plate dispersion relation. Also, the influence of the plate’s motion on wave propagation in the open water field and within the gap of the ring is studied. Numerical results are presented for illustrative purposes.     

Eccentricity effects on acoustic radiation from a spherical source suspended within a thermoviscous fluid sphere

Acoustic radiation from a spherical source undergoing angularly periodic axisymmetric harmonic surface vibrations while eccentrically suspended within a thermoviscous fluid sphere, which is immersed in a viscous thermally conducting unbounded fluid medium, is analyzed in an exact fashion. The formulation uses the appropriate wave-harmonic field expansions along with the translational addition theorem for spherical wave functions and the relevant boundary conditions to develop a closed-form solution in form of infinite series. The analytical results are illustrated with a numerical example in which the vibrating source is eccentrically positioned within a chemical fluid sphere submerged in water. The modal acoustic radiation impedance load on the source and the radiated far-field pressure are evaluated and discussed for representative values of the parameters characterizing the system. The proposed model can lead to a better understanding of dynamic response of an underwater acoustic lens. It is equally applicable in miniature transducer analysis and design with applications in medical ultrasonics.     

Probabilistic characterization of the effect of transient stochastic loads on the fatigue-crack nucleation time

The rainflow counting algorithm for material fatigue is both simple to implement and extraordinarily successful for predicting material failure times. However, it neglects memory effects and time-ordering dependence, and therefore runs into difficulties dealing with highly intermittent or transient stochastic loads with heavy tailed distributions. Such loads appear frequently in a wide range of applications in ocean and mechanical engineering, such as wind turbines and offshore structures. In this work we employ the Serebrinsky–Ortiz cohesive envelope model for material fatigue to characterize the effects of load intermittency on the fatigue-crack nucleation time. We first formulate efficient numerical integration schemes, which allow for the direct characterization of the fatigue life in terms of any given load time-series. Subsequently, we consider the case of stochastic intermittent loads with given statistical characteristics. To overcome the need for expensive Monte-Carlo simulations, we formulate the fatigue life as an up-crossing problem of the coherent envelope. Assuming statistical independence for the large intermittent spikes and using probabilistic arguments we derive closed expressions for the up-crossing properties of the coherent envelope and obtain analytical approximations for the probability mass function of the failure time. The analytical expressions are derived directly in terms of the probability density function of the load, as well as the coherent envelope. We examine the accuracy of the analytical approximations and compare the predicted failure time with the standard rainflow algorithm for various loads. Finally, we use the analytical expressions to examine the robustness of the derived probability distribution for the failure time with respect to the coherent envelope geometrical properties.