A connection between linear normal modes and lines of curvature on the potential surface

For the N-degree-of-freedom of linear conservative vibratory systems, the corresponding potential functions can be viewed as N-hypersurfaces in (N + 1)-dimensional space. In this paper, a connection between the geometrical properties (principal curvatures, curvature lines) of potential surfaces and the vibratory characteristics (natural frequencies, linear modes) of the system is built. It is proved that the linear normal modes are exactly the projections of the lines of curvature on the potential surface onto the configuration space with metric m_ij (the mass-matrix); and that the squared natural frequencies are exactly the principal curvatures, at the origin of the configuration space, of the potential surface.     

Effects of the curvature radius of a cylindrical shell on the stress intensity factors of surface cracks

The effects of the curvature radius of a cylindrical shell on stress intensity factors are investigated in circumferential (inner and outer) semielliptical surface cracks in a cylindrical shell. What is new in this paper is to have given: (1) The stress intensity factors for surface cracks in a cylindircal shell are determined by photoelastic technique. (2) By a special method photoelastic slices are handled for obtaining a clear caustic curve, and the stress intensity factors for surface cracks in a cylindrical shell are determined by the caustic method. (3) An approximate equation of curvature correction factor F_c is proposed. (4) Effects of the curvature radius R of a cylindrical shell on the stress intensity factors of surface cracks are obtained. The results of this paper are in fair agreement with already existing analytical results. The approximate equation of curvature correction factor F_c can be widely used for engineering purposes.     

常曲率弯道二维流动稳定性与非线性演化规律分析

国内外对弯曲河流的研究从线性特性转移到不稳定性与非线性特性上来, 弯曲河流作为一个不稳定的系统, 主要是受到水流和边界的不稳定性的影响.针对弯曲河流中存在的水流动力的不稳定性与非线性特征,利用弱非线性理论与摄动法进行展开, 在微弯条件下建立了时间模式下的常曲率 弯道二维水流扰动幅值与扰动幅角的非线性演化方程.研究了在不同弯曲度下的扰动发展特征,探讨了扰动流场在弯曲度影响下的时空分布的规律,分析了扰动流场中出现的扰动漩涡的运动过程, 阐述了弯曲度对弯曲边界内水流不稳定性的影响, 具体表现为在微弯的情况下, 弯道的弯曲度增大会提高河湾内水流的稳定性, 扰动振幅与扰动幅角会随弯曲度的增大其衰减速度更快, 并且扰动流速的对称性在弯曲度较大时减弱, 逐渐向弯道的凸岸偏斜, 在中性状态附近 的弯道水流动力具有对流不稳定性和非线性衰减的特征.本文的研究成果为构建水流动力非线性与床面形态几何非线性相互作用的模型提供了思路.      

灰色相关性分析在结构静力损伤识别中的应用

基于灰色理论的相关性分析方法，首次提出了灰色曲率关联系数的概念并将其应用到结构的静力损伤识别中，提出了对局部损伤十分敏感的静态位移曲率置信因子SDCACi，通过该因子的大小对各节点所连接的单元是否会发生损伤进行精确的判断，然后运用最小二乘法对损伤区域的损伤程度进行识别．并将该方法应用于两端固支梁的损伤识别中，由识别结果可以证明：不论测量数据(用有限元仿真计算并考虑了测量误差)的多少，该方法对结构中的单损伤和多损伤都能进行准确的定位，因此该方法在大型结构及复杂结构的损伤识别中具有广阔的应用前景．     

Bifurcations of Umbilic Points and Related Principal Cycles

The simplest patterns of qualitative changes on the configurations of lines of principal curvature around umbilic points on surfaces whose immersions into ³ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper.Global effects, due to umbilic bifurcations, on these configurations such as the appearance and annihilation of periodic principal lines, called also principal cycles, are also studied here.     

变曲率连续滚滑接触安定性的数值方法研究

针对接触表面变曲率的特点,引入局部坐标系,构造出局部坐标下残余应力应变场的分布状态,建立了变曲率连续啮合过程中安定状态残余应力的计算方法。该数值方法将弹塑性问题分解为弹性问题和特征应变决定的残余问题,并采用增量映射方法求解特征应变决定的残余问题,可直接得到接触安定状态下的接触残余应力,并随之进行安定极限的判定。采用该数值方法计算了不同曲率处接触点的安定极限,给出了安定极限与摩擦因数之间的关系,并与有关数值结果相比较,验证了该算法的有效性。     

Numerical prediction of a turbulent curved wake and comparison with experimental data

Numerical studies of the curved wake of a NACA 0012 airfoil of chord length 0.150 m are presented. The airfoil is placed in air at 10 m/s in the straight section of a duct of 0.457 m × 0.457 m cross‐section followed by a 90° bend with a mean radius‐to‐height ratio of 1.17. The trailing edge is located at one chord length upstream of the bend entry plane. The authors' own measurements are used to define the boundary conditions and for comparison with the predicted results. The numerical models are based on the time‐averaged, three‐dimensional conservation equations of fluid flow, incorporating the k–ε, RNG k–ε, realizable k–ε and the Reynolds stress turbulence models. The results show that the models are capable of predicting the effects of curvature on the wake development. However, quantitative differences between prediction and experiment exist. The results obtained using the Reynolds stress model show better agreement with the experimental data, compared with the k–ε based models, but not consistently for all parameters. There are also better predictions by the RNG k–ε and realizable k–ε models compared with the standard k–ε model. The predicted results using the RNG k–ε are closer to experimental data than the realizable k–ε. Copyright © 2005 John Wiley & Sons, Ltd.     

用白光光源获得斜率和曲率的干涉法

本文介绍了用白光光源和衍射光栅作为基本光学元件,在滤波光路中获得物面斜率和曲率条纹的实验原理和方法.讨论了该方法的一些基本性质的和特点.     

Combined free and forced convection for developed flow in curved pipes with finite curvature ratio

Combined free and forced convection for developed flow in a curved pipe with arbitrary curvature ratio is studied numerically. The curved pipe is heated with axially uniform heat flux, while the wall temperature is maintained peripherally uniform. The buoyancy force is accounted by the Boussinesq approximation. The effects of the Dean, Prandtl, and Rayleigh numbers and especially of a wide range of curvature ratios on the flow resistance and the average heat transfer rate are presented. The significant distortion of the dividing streamline and the appearance of the secondary flow with one dominant cell for pipe flow with higher buoyancy force and curvature ratio are also discussed.     

Simulating turbulent Dean flow in Cartesian coordinates

A simplified approach to simulate turbulent flows in curved channels is proposed. A set of governing equations of motion in Cartesian coordinates is derived from the full Navier–Stokes equations in cylindrical coordinates. Terms to first order in the dimensionless curvature parameter are retained, whereas higher‐order terms are neglected. The curvature terms are implemented in a conventional Navier–Stokes code using Cartesian coordinates. Direct numerical simulations (DNS) of turbulent flow in weakly curved channels are performed. The pronounced asymmetries in the mean flow and the turbulence statistics observed in earlier DNS studies are faithfully reproduced by the present simplified Navier–Stokes model. It is particularly rewarding that also distinct pairs of counter‐rotating streamwise‐oriented vortices are embedded in the simulated flow field. Copyright © 2008 John Wiley & Sons, Ltd.     

Near-Equidiffusion Disturbed Premixed Flames with Wider Reaction Zones

Most of the asymptotic considerations of the interaction of premixed flames with a general flow, i.e. curved and stretched flames subjected to time dependent flow, are dedicated to high activation energy asymptotes. Therefore, in these considerations the reaction zone is thin and the temperature within the reaction zone is constant to the leading order of approximation. Here we consider an order unity activation energy for near-equidiffusion flames and show that the flame speed relations obtained are distinct from those obtained by high activation energy asymptotes. The flame is assumed to be thin in comparison with the flow scales but the reaction zone is no longer thin in comparison with the flame width. Although obtaining analytical solutions is problematic even for undisturbed flames with wide reaction zones, we found that the propagation speed of disturbed premixed flames with wide reaction zones is determined by analytical integrals involving the temperature profile of the undisturbed flame. We also found independent effects of curvature and stretch for the flames with wider reaction zones. This revised version was published online in July 2006 with corrections to the Cover Date.     

Conical Mach reflection of moving shock waves

Conical Mach reflections differ from those of the equivalent plane, two-dimensional Mach reflection because in axisymmetry, the disturbances generated at the reflecting surface are modified by their more rapidly increasing or decreasing area as they move towards or away from the centerline. Equations for conical Mach reflection cases have now been developed using a simplified ray-shock theory formulation based on the initial assumption that the stem is straight and normal to the wall. These are in a form that applies generally. Their simple structure provides an easy conceptual understanding of self-similarity and non-self-similarity as well as a clear mathematical approach for the development of the curved triple-point locus of the latter by integration. They provide a quick and direct solution in all cases and can easily incorporate the Mach stem curvature by progressively calculating the new ray direction. A range of cases has been considered and results are presented for converging and diverging, self-similar and non-self-similar cases.     

Inlet angle distribution of last stage moving blades for large steam turbines

A simplified analytical method is presented for use in preliminary design studies on the final stages of large steam turbines to derive the required moving blade twist distribution to achieve near-zero incidence. The method incorporates the influence of nozzle twist and of casing flare which is shown to enter the analysis mainly through its effect on the distribution of streamline curvature. Provided flare is properly incorporated into the design, there appears to be no inherent reason why higher flare should limit performance unduly. Optimisation of moving blade twist for variations in load and back pressure is also investigated. Blade skeletons derived analytically are subjected to full throughflow calculations and refined where necessary to improve their likely performance. The results are compared with results of calculations with published commercial blade angle distributions, designed for both low- and high-flare turbines. The results of the throughflow calculations on the blades derived here, and the comparisons with apparently successful commercial blades, suggest that this method can be used as a preliminary design tool, freeing the designer from repetitive trial runs with a full throughflow calculation.     

曲率半径对前缘气动热与结构响应的影响

针对高超声速飞行器铌合金前缘结构, 研究了不同曲率半径对前缘结构 温度场、应力场和变形场的影响. 首先建立高超声速气动加热模型, 采用有限体积法得到热 环境参数, 并运用有限元法计算结构的温度、应力和变形. 结果表明: 不同时间的温度场分 布和曲率半径密切相关, 温升过程中应力最大值出现在曲率半径为1\,mm时; 随时间推移, 曲 率半径越大应力越低; 而位移随曲率半径的增加而增大.     

Single-beam shearing Moiré interferometry for measuring slope and curvature of bent plates

The single-beam shearing interferometry with high-frequency phase-type blazed grating for measuring the slope and curvature contour fringes of bending plates is presented. As it is characterized by excellent fringe contrast, high sensitivity and extensive range, this method is effective for plate bending problems. Two examples of application to the clamped circular bent plate and the rectangular bent plate with edge notch are given.     

Free vibrations of rectangular orthotropic shallow shells with varying thickness

The paper proposes a numerical-analytic approach to studying the free vibrations of orthotropic shallow shells with double curvature and rectangular planform. The approach is based on the spline-approximation of unknown functions. Calculations are carried out for different types of boundary conditions. The influence of the mid-surface curvature and variable thickness on the behavior of dynamic characteristics is studied __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 102–115, June 2007.     

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利用自由液面曲率参量，在传统求解方法基础上对相应结点网格速度引入修正，以提高计算精度.域内网格速度计算，由传统线性关系改用分段函数表达，以保证网格剖分的质量. 将二维矩形贮箱内流体晃动问题作为算例，求解激励响应结果并与已有物理实验进行对比,二者符合较好.     

A modified STRUCT model for efficient engineering computations of turbulent flows in hydro-energy machinery

A modified STRUCT (MST) turbulence model for efficient engineering computations of turbulent flows in hydro-energy machinery is proposed in this paper. The MST model switches between URANS and LES-like modes using a new damping function to adjust the turbulent viscosity. Compared with the original STRUCT method, the modifications are as follows: (1) the BSL k-ω model with the Spalart-Shur correction is chosen as the new baseline to improve the sensitivity to rotation and curvature; (2) a new adaptive time-scale ratio is proposed to avoid the arbitrariness of geometric averaging operation in the original method; (3) the normalized helicity is introduced into the new damping function to detect the energy backscatter phenomenon. Five classical high Reynolds number flow cases are tested. The results show that the turbulent viscosity of the MST model is reasonably reduced in the massively separated regions and LES-like mode is activated, which captures more turbulent vortices and fluctuations on the URANS grids. With high efficiency and robustness, the MST model inherits the advantages of the original STRUCT method and improves the prediction accuracy of the turbulence with rotation and curvature, which enables efficient engineering computations of turbulent flows in hydro-energy machinery.     

重力作用下液体自由面的数值解

给出一种重力作用下静止液体自由面的数值解方法。根据流体力学Young-Laplace公式和微分几何学主曲率方程,并利用液体自由面的对称性质,导出重力作用下静止液体自由面母线的常微分控制方程。为了使该方程便于利用现有的Runge-Kutta法,根据问题的物理背景,将体积边界条件转化为固-液接触面半径条件;采用一种坐标变换,避免了该方程数值计算中刚性问题的出现。最后给出了算例。