Propagation of a long wave in a curved duct (I) basic analysis of long wave propagation in a curved duct with variable cross section

The propagation of a long wave in a three-dimensional curved duct with variable cross section is studied in this paper. It is shown that a three-dimensional Helmholtz equation can be decomposed into a two-dimensional Laplace (or Poisson) equation and a one-dimensional Webster equation by the curvilinear orthogonal coordinate system, non-dimensionization of reduced wave equation and regular perturbation with small parameterka, wherek is the wave number anda is the characteristic radius of the duct. The influences of the duct's geometric parameters (the area variation of the cross section, the curvature and torsion of the central line) on the asymptotic expansion of the solution are analysed. It is concluded that the effects of the variation of the cross sectional area first appear in the first term of the asymptotic expansion, and when the cross section shape has certain symmetric properties, the effects of the curvature and torsion of the central line first appear in the third and the fourth terms, respectively. An example of long wave propagation in a curved circular duct is also given at the end of this paper.     

Modes of vortex formation and transition to three-dimensionality in the wake of a freely vibrating cylinder

The present work is aimed to give some insight into the relation between vortex shedding modes and transition to three-dimensionality in the wake of a freely vibrating cylinder by establishing a numerical model and analyzing the relevant results of two- and three-dimensional simulations. The compressible flow past an elastically-mounted cylinder is solved by using the immersed boundary method (IB method). The cylinder is free to vibrate in the transverse direction with zero structure damping. The response of displacement amplitude is studied with the variation of reduced velocity. Whether P+S mode exists in three-dimensional flow and the occurrence of 2P mode is caused by flow transition from two-dimensional to three-dimensional are problems of concern. Both 2P and P+S wake modes are observed in two- and three-dimensional simulations. The numerical results indicate that the flow transition from two-dimensional to three-dimensional is coupled with the cylinder vibration in the synchronization/lock-in regime. The wake formation given by three-dimensional simulations suggests that the P+S mode might exist in reality when the flow is reverted to two-dimensional by vortex induced vibration (VIV) at Re=300–350. When Reynolds number increases to 425, the wake formation undergoes transition to three-dimensionality and 2P mode is observed. The effect of mass ratio on the flow transition to three-dimensionality is studied. The relationship between wake modes and aerodynamic forces is discussed.     

Effect of Surface Corrugation on Convection in a Three-Dimensional Finite Box of Fluid Saturated Porous Material

The problem of finite amplitude thermal convection in a three-dimensional finite box of fluid saturated porous material is investigated, when the lower boundary of the fluid is corrugated. The nonlinear problem of three-dimensional convection in the box for the values of the Rayleigh number close to the classical critical value and for small values of the amplitude of the corrugations is solved by a perturbation technique. The preferred mode of convection is determined by stability analysis. In the absence of corrugation three-dimensional modes of convection can be either stable or unstable depending on the values of the aspect ratios of the box, while two-dimensional rolls are always stable, provided that the box aspect ratios allow the existence of such modes of convection. In the presence of boundary corrugation with the appropriate form, different three-dimensional or two-dimensional modes of corrugation can be stable or unstable. For a rough boundary with local roughness sites, the location, size, and number of the roughness elements plus the wave numbers of the convection modes and the box aspect ratios can all play a role leading to either stable or unstable particular three- or two-dimensional flow patterns. For a wavy boundary, resonant wave-vector excitation can lead to the preference of stable two- or three-dimensional flow patterns whose wave vectors are in a subset of those due to the wavy boundary, while nonresonant wave-vector excitation can lead to the preference of stable flow patterns whose wave vectors are not generally in a subset of those due to the wavy boundary. Heat transported by convection can either be enhanced or be reduced by certain proper forms of the corrugations and by appropriate values of the box aspect ratios. Due to the surface corrugation highly subcritical modes of convection are stable, while highly supercritical modes of convection are unstable. Received 24 July 1998 and accepted 11 April 1999     

Effect of vertical heat transfer on thermocapillary convection in an open shallow rectangular cavity

In order to understand the effect of the vertical heat transfer on thermocapillary convection characteristics in a differentially heated open shallow rectangular cavity, a series of two- and three-dimensional numerical simulations were carried out by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and the aspect ratio ranged from 12 to 30. Results show that thermocapillary convection is stable at a small Marangoni number. With the increase of the heat flux on the bottom surface, thermocapillary convection transits to the asymmetrical bi-cellular pattern with the opposite rotation direction. The roll near the hot wall shrinks as the Marangoni number increases. At a large Marangoni number, numerical simulations predict two types of the oscillatory thermocapillary flow. One is the hydrothermal wave, which is dominant only in a thin cavity. The other appears in a deeper cavity and is characterized by oscillating multi-cellular flow. The critical Marangoni number for the onset of the oscillatory flow increases first and then decreases with the increase of the vertical heat flux. The three-dimensional numerical simulation can predict the propagating direction of the hydrothermal wave. The velocity and temperature fields obtained by three-dimensional simulation in the meridian plane are very close to those obtained by two-dimensional simulation.     

On waves due to rolling of a ship in water of finite depth

Summary The problem of the generation of waves due to small rolling oscillations of a thin vertical plate partially immersed in uniform finite-depth water is investigated here by utilizing two mathematical methods assuming the linearised theory of water waves. In the first method, the use of eigenfunction expansion of the velocity potentials on the two sides of the plate produces the amplitude of wave motion at infinity in terms of an integral involving the unknown horizontal velocity across the gap, and also in terms of another integral involving the unknown difference of the potential across the plate. These unknown functions satisfy two integral equations. Any one of these, when solved numerically, can be used to compute the amplitude of the wave motion set up at either infinity on the two sides of the plate for various values of the wave number.In the second method, the problem is formulated in terms of a hypersingular integral equation involving the difference of the potential function across the plate. The hypersingular integral equation is solved numerically, and its numerical solution is used to compute the wave amplitude at infinity. The two methods produce almost the same numerical results. The results are illustrated graphically, and a comparison is made with the deep-water result. It is observed that the deep-water result effectively holds good if the plate is partially immersed to the order of one-tenth of the bottom depth.This work was initiated when the first Author was visiting Mathematics Department, Indian Institute of Science, Bangalore. It was partially supported by DST, and by CSIR. The authors take this opportunity to thank the Managing Editor for his suggestions to improve the paper in the present form.     

流体动力干扰对单排圆柱桩列波浪力的影响

多物体之间的流体动力干扰特性对超大型海洋结构物的设计和研究十分重要用波动源在截面周线上分布的方法，就垂直桩柱间三维流体动力干扰对波浪力的影响进行了系统的研究，桩柱的数目可达１００余根得到了柱间流体动力干扰力学机理的若干新的特性尤应指出的是，当桩柱根数超过某一数量后，桩柱上的受力表现出有规律的连续依赖性当柱数很大时，无论柱数是奇数还是偶数，中间大部分的桩柱都将表现出均匀的受力特征这些特性的发现对其他形式多体结构物流体动力干扰的研究也有重要的借鉴和指导意义     

Optimizing thread-root contours using photoelastic methods

The research work described in this paper was part of a program which was conducted to determine stress distributions in a complex multicomponent pressure vessel and in several redesigned versions of the same vessel. Three-dimensional photoelasticity methods were selected for the analyses when it was determined that a number of highly stressed internal areas were inaccessible for evaluation of stresses by electrical-resistance strain gages and uncertainties regarding load transfer between mating components made an analytical determination impractical. The results of the stress determinations in the original vessel indicated that the highest stresses were located in the small fillets at the roots of the sectored threads which were used to permit quick and convenient assembly of the parts. Results of a prototype fatigue study verified the importance of these stresses when it was determined that fracture initiated in this high-stress area. Because of design specifications and the nature of conditions under which the vessel operates, stress reductions had to be accomplished using thread form or rootcontour changes. Since three-dimensional studies are time consuming and expensive, two-dimensional methods were selected to evaluate a number of changes before final evaluation was made in a three-dimensional model. Comparisons between results from the two- and three-dimensional studies indicated that the percent improvements indicated in the two-dimensional studies were roughly obtained in the three-dimensional vessels.     

梯形应力脉冲在弹性杆中的传播过程和几何弥散

在应力波传播过程中,几何弥散效应往往难以避免.对应力波在弹性杆中传播的几何弥散效应进行解析分析,对于基础波动问题研究以及材料动态力学行为表征等课题,显得至关重要.本文简单说明了弹性杆中考虑横向惯性修正的一维 Rayleigh-Love应力波理论,概述了其波动控制方程的变分法推导过程;针对 Hopkinson杆实验中常用的梯形应力加载脉冲,建立了相应的偏微分方程初边值问题的求解模型,并运用 Laplace变换方法研究了脉冲在杆中传播的几何弥散现象;根据留数定理进行 Laplace反变换,给出了杆中不同位置和时刻的应力波的级数形式解析解,分析了计算项数对结果收敛性的影响;将解析计算结果与采用三维有限元数值模拟的计算结果进行对比,两者吻合程度良好,从而证明 Rayleigh-Love横向惯性修正理论可以有效地表征典型 Hopkinson杆实验中的几何弥散效应.在此基础上围绕梯形加载脉冲的弥散效应进行参数研究,定量描述了传播距离、泊松比、脉冲斜率等参数的影响.本文给出的 Rayleigh-Love杆在梯形加载条件下的解析解,揭示了几何弥散效应的本质规律,可以用于实际实验的弥散修正过程.      

非定常激波三维双楔面反射的数值研究

采用基于MUSCL-Hancock插值的有限体积方法, 在非结构自适应网格上求解三维Euler方程, 研究了非定常激波在三维垂直双楔面上的反射现象. 研究结果表明, 由于三维效应的影响, 通过二维非定常激波反射理论预测的三维马赫反射区域范围存在着一定的局限, 在该区域范围内出现了一种非典型的三维突起结构, 即第二类三维马赫干. 另外, 对于不同的激波马赫数和楔面倾角组合, 非定常激波在三维双楔面上会形成具有四波结构的三维马赫反射或具有三波结构的三维规则反射.      

Effect of Maxwell stress on a moving crack with polarization saturation region in ferroelectric solid

The focus of this work is on a generalized two-dimensional problem of a crack moving in a piezoelectric solid subjected to uniform electrical load at infinity. The novel point includes that the electric field inside the crack is taken into account when polarization saturation region exists. Based on the extended Stroh formalism and complex function method, explicit expressions of both the stress fields in the solid and electric fields inside the crack are derived by using semi-permeable crack model, respectively. Effect of Maxwell stress along the crack surface is investigated and the results are illustrated graphically. It is shown that the moving speed of the crack cannot exceed the lowest bulk wave speed. It is also found that the medium properties inside the crack and surrounding the ferroelectric solid at infinity directly affect the Maxwell stress, and as a result the Maxwell stresses are remarkable and cannot be ignored under different electric load.     

倾斜界面耦合弹性层中的渡越辐射能

物理学中,摄动源在非均匀介质中或非均匀介质附近匀速直线运动所产生的能量辐射现象称为渡越辐射.列车沿轨道运行,由轮轨接触产生的弹性波在非均匀轨道和基础中传播将发生渡越辐射,而轨道和基础的非均匀性集中体现在不同轨道基础之间的过渡段(如路桥过渡段、桥隧过渡段或有砟-无砟轨道过渡段).为研究车致弹性波在过渡段中引发的渡越辐射现象,本文以典型高速铁路路桥过渡段结构形式为依据,建立了二维平面应力渡越辐射能计算模型.其中,两个材料参数不同的半无限弹性层由一倾斜界面耦合,底端固定,上表面自由,一个集中载荷在自由表面上匀速运动.界面两侧弹性体中的波动方程均分解为本征场、自由场两个部分分别求解,其中自由场波动方程采用分离变量法数值求解.通过模型求解得到了不同载荷移动速度和界面倾斜角度条件下的渡越辐射能及界面附近应变能密度.结果表明,渡越辐射能的大小随载荷移动速度增大单调非线性增大,移动载荷速度达到刚度较大一侧介质表面波速的74%时产生的渡越辐射能就将超过载荷本身激发的本征场应变能;界面倾斜角度越大,即两侧介质刚度过渡距离越短,渡越辐射能与本征场应变能比值越大.      

Wave resistance of a viscous liquid

In this paper we examine the resistance encountered by a system of normal stresses during its rectilinear motion along the surface of a viscous liquid of infinite depth. The problem is solved in the linear formulation, i.e., it is assumed that amplitudes of the waves which arise are small and the waves are shallow. The solution for the two-and three-dimensional problems is obtained in the general case in closed form. In the two-dimensional case a detailed study is made of the case when a constant pressure p₀, moving with the constant velocity U, is given on a segment of length 2l. In the three-dimen-sional problem the case is studied when the normal stress is concentrated on a segment of a straight line of length 2l, which can replace a ship moving along a straight course with the constant velocity U. The integrals obtained in both cases are studied using the stationary phase method, the application of which for the three-dimensional integrals with respect to a volume with boundaries is justified in §1 of the paper. As a result we obtain equations for the wave resistance in the two- (§2) and three-dimensional (§3) cases.     

An experimental study of plane bubbles rising at inclination

The characteristics of bubbles rising in two-dimensional tanks is investigated. It is found that the influence of fluid properties, while negligible for vertically rising bubbles, measurably affects the inclined bubble rise velocity. The measured increase in the rise velocity at inclination relative to the vertical value is explained qualitatively. In addition, a comparison is made between the wave analogy correlation of Maneri & Mendelson and vertical rise velocity data. The comparisons show that the correlation predicts the three-dimensional effects of tank spacing when the data are reduced on an equivalent bubble radius basis.     

Nonlinear effects caused by pulsed periodic supply of energy in the vicinity of a symmetric airfoil in a transonic flow

Changes in the structure of a transonic flow around a symmetric airfoil and a decrease in the wave drag of the latter, depending on the energy-supply period and on localization and shape of the energy-supply zone, are considered by means of the numerical solution of two-dimensional unsteady equations of gas dynamics. Energy addition to the gas ahead of the closing shock wave in an immediate vicinity of the contour in zones extended along the contour is found to significantly reduce the wave drag of the airfoil. The nature of this decrease in drag is clarified. The existence of a limiting frequency of energy supply is found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 64–71, May–June, 2006.     

Numerical study on the dynamics of a two-raft wave energy conversion device

This paper presents a dynamic analysis of a two-raft wave energy conversion device based on the three-dimensional wave radiation-diffraction method. The device consists of two hinged cylindrical rafts of elliptical cross section and a power take-off system at the joint. The effect of raft length, linear damping and spring coefficient in the power take off (PTO) system, axis ratio (ratio of minor axis to major axis of raft elliptical cross section) and raft radius of gyration on wave energy capture factor has been investigated in frequency domain, while the effects of a nonlinear Coulomb power take-off, raft radius of gyration and latching control have been studied in time domain. The difference in the performance of a raft-typed device obtained using a linear damping and a Coulomb damping is also illustrated.     

特征线法在三维弹／粘塑性波数值解析中的应用

根据广义特征理论建立了求解三维非线性应力波传播问题的特征关系式，并采用特征线法对纵向冲击载荷下弹／粘塑性矩形截面棒的三维应力波传播过程进行了数值模拟。     

Traveling gravity water waves in two and three dimensions

This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.     

Some aspects of streamwise vortex behavior during oblique shock wave/vortex interaction

An experimental study of the flowfield generated by the interaction of a streamwise vortex having a strong wake-type axial Mach number profile and a two-dimensional oblique shock wave was conducted in a Mach 2.49 flow. The experiments were aimed at investigating the dynamics of supersonic vortex distortion and to study downstream behavior of a streamwise vortex during a strong shock wave/vortex encounter. The experiments involved positioning an oblique shock generator in the form of a two-dimensional wedge downstream of a semi-span, vortex generator wing section so that the wing-tip vortex interacted with the otherwise planar oblique shock wave. Planar laser sheet visualizations of the flowfield indicated an expansion of the vortex core in crossing a spherically blunt-nose shock front. The maximum vortex core diameter occurred at a distance of 12.7 mm downstream of the wedge leading edge where the vortex had a core diameter of more than double its undisturbed value. At distances further downstream the vortex core diameter remained nearly constant, while it appeared to become more diffused at distances far from the wedge leading edge. Measurements of vortex trajectory revealed that the vortex convected in the freestream direction immediately downstream of the bulged-forward shock structure, while it traveled parallel to the wedge surface at distances further downstream. The turbulent distorted vortex structure which formed as a result of the interaction, was found to be sensitive to downstream disturbances in a manner consistent with incompressible vortex breakdown. Physical arguments are presented to relate behavior of streamwise vortices during oblique and normal shock wave interactions. Received 7 September 1996 / Accepted 10 February 1998