带裂纹矩形板自由振动解析解

选取带有补充项的双重正弦傅里叶级数作为振型函数通解,来解析研究带裂纹矩形板的自由振动特性。先将带裂纹矩形板分割成若干小矩形板,利用各小矩形板的边界条件,并结合振型函数中待定常数的物理意义,简化得到各小矩形板的振型函数,再结合各板的控制方程、未使用的边界条件、公共边协调条件及本文提出公共自由角点的协调条件,建立求解频率的代数方程组,然后将其转化为广义特征值问题来求解带裂纹矩形板的无量纲频率;最后选取具体参数进行计算并与文献结果对比,吻合良好,证明了本文采用的研究方法以及所提出公共角点协调条件的正确性和合理性。由于该振型函数能满足矩形板的任意边界约束,且其中的待定常数具有明确的物理意义,所以可使矩形板问题的求解统一化、简单化和规律化。     

Analytical techniques for analysis of fully developed laminar flow through rectangular channels

Approximate analytical techniques are established to determine the velocity distribution for laminar fluid flow through straight rectangular channels. The separation of variables is adopted for the exact analytical solution. The closed form solution for the temperature distribution in the flow field is demonstrated and the Nusselt number is evaluated. The result determined by different techniques has been presented in a comparative way to easily understand the accuracy level of predictions.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper,a new analytical method of symplectic system.Hamiltonian system,is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain.In the system,the fundamental problem is reduced to all eigenvalue and eigensolution problem.The solution and boundary conditions call be expanded by eigensolutions using ad.ioint relationships of the symplectic ortho-normalization between the eigensolutions.A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space.The results show that fundamental flows can be described by zero eigenvalue eigensolutions,and local effects by nonzero eigenvalue eigensolutions.Numerical examples give various flows in a rectangular domain and show effectivenees of the method for solving a variety of problems.Meanwhile.the method can be used in solving other problems.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.     

Forced convection gaseous slip flow in circular porous micro-channels

Laminar forced convection of gaseous slip flow in a circular micro-channel filled with porous media under local thermal equilibrium condition is studied numerically using the finite difference technique. Hydrodynamically fully developed flow is considered and the Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous domain. The present study reports the effect of several operating parameters (Knudsen number (Kn), Darcy number (Da), Forchhiemer number (Γ), and modified Reynolds number ) on the velocity slip and temperature jump at the wall. Results are given in terms of the velocity distribution, temperature distribution, skin friction , and the Nusselt number (Nu). It is found that the skin friction is increased by (1) decreasing Knudsen number, (2) increasing Darcy number, and (3) decreasing Forchheimer number. Heat transfer is found to (1) decrease as the Knudsen number, or Forchheimer number increase, (2) increase as the Peclet number or Darcy number increase.     

Simulation of flow boiling in micro-channels: Effects of inlet flow rate and hot-spots

This paper presents the findings of a numerical study on the flow boiling in a micro-channel heat sink. The Navier-Stokes equations, energy equation, and the continuity equation are solved in a finite-volume framework using the front-tracking method. The numerical method is validated by comparison with the experimental results for a slug bubble growth, and vertical flow boiling. The numerical method is then used to study the effect of changing the inflow mass-velocity on the heat transfer coefficient, bubble size distribution, and the bubble nucleation frequency for a constant heat flux. The mean heat transfer coefficient of all the cases is found to be nearly twice that of the single-phase heat transfer coefficient. The bubble nucleation frequency is found to increase monotonically with the inflow mass-velocity. The bubble size distribution along the channel is found to become flatter as the mass-velocity is increased. We identify three distinct phases of the bubble evolution, namely the initial rapid growth phase, the boiling dominant phase, and finally the condensation dominant phase. Subsequently, the numerical method is used to study the effect of having a hot-spot near the bubble nucleation site on the heat transfer characteristics. It is found that the bubble nucleation frequency increases and the bubbles’ maximum volume decreases as the intensity of the hot-spot is increased for a fixed inlet flow rate. It is also observed that the average heat transfer coefficient does not change significantly with changing the intensity of the hot-spot, and that the bubble size distribution along the channel becomes flatter as the intensity of the hot-spot is increased.     

Variational iteration method for two-dimensional steady slip flow in micro-channels

In this paper, the two-dimensional steady slip flow in microchannels is investigated. Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). The Navier-Stokes equations for two-dimensional steady slip flow in microchannels are reduced to a nonlinear third-order differential equation by using similarity solution. The variational iteration method (VIM) is used to solve this nonlinear equation analytically. Comparison of the result obtained by the present method with numerical solution reveals that the accuracy and fast convergence of the new method.     

Analytical solutions for transverse distributions of stream-wise velocity in turbulent flow in rectangular channel with partial vegetation

The theory of poroelasticity is introduced to study the hydraulic properties of the steady uniform turbulent flow in a partially vegetated rectangular channel. Plants are assumed as immovable media. The resistance caused by vegetation is expressed by the theory of poroelasticity. Considering the influence of a secondary flow, the momentum equation can be simplified. The momentum equation is nondimensionalized to obtain a smooth solution for the lateral distribution of the longitudinal velocity. To verify the model, an acoustic Doppler velocimeter (ADV) is used to measure the velocity field in a rectangular open channel partially with emergent artificial rigid vegetation. Comparisons between the measured data and the computed results show that the method can predict the transverse distributions of stream-wise velocities in turbulent flows in a rectangular channel with partial vegetation.     

Experimental study of flow and heat transfer in rib-roughened rectangular channels

Secondary flow patterns, pressure drop and heat transfer in rib-roughened rectangular channels have been investigated. The aspect ratio of the channels is 1–8, and ribs are attached to the wide channel walls in order to set up swirling motions. The geometries tested consist of channels having cross ribs, parallel ribs, cross V-ribs, parallel V-ribs, and multiple V-ribs (Swirl Flow Tube). The flow patterns were investigated using smoke wire visualization and LDV measurements. The smoke wire experiments have been performed at Re=1100 and the LDV measurements at Re=3000 at periodic fully developed conditions. The heat transfer and pressure drop are described by j and f factors for Reynolds numbers from 500 to 15 000. The distributions of axial mean velocity and turbulent fluctuations are strongly influenced by the secondary flows. Large mean velocities and small fluctuations are found in regions where the secondary flow is directed towards a surface, while small mean velocities and large fluctuations are found in regions where the secondary flow is directed away from a surface. The Swirl Flow Tube provides a significant increase in the j factor at Reynolds numbers from 1000 to 2000, but unfortunately also an increase in the f factor. At higher Reynolds numbers, the j and f factors of the Swirl Flow Tube are of the same order of magnitude as for the other rib-roughened channels. It is found that the flow direction in a channel with parallel V-ribs has important influence on the j/f ratio. At Reynolds numbers above 4000, this channel provides the highest j/f ratio if the V-ribs are pointing upstream; while it provides the lowest j/f ratio of all rib configurations, if the V-ribs are pointing downstream.     

Large-scale flow structures and heat transport of turbulent forced and mixed convection in a closed rectangular cavity

Results of an experimental study of flow structure formation and heat transport in turbulent forced and mixed convection are presented. The experiments were conducted in a rectangular cavity with a square cross section, which has an aspect ratio between length and height of Γ_xz = 5. Air at atmospheric pressure was used as working fluid. The air inflow was supplied through a slot below the ceiling, while exhausting was provided by another slot, which is located directly above the floor. Both vents extend over the whole length of the cell. In order to induce thermal convection the bottom of the cell is heated while the ceiling is maintained at a constant temperature. This configuration allows to generate and study mixed convection under well defined conditions. Results of forced convection at Re = 1.07 × 10⁴ as well as mixed convection at 1.01 × 10⁴ ? Re ? 3.4 × 10⁴ and Ra = 2.4 × 10⁸ (3.3 ? Ar ? 0.3), which were obtained by means of Particle Image Velocimetry and local temperature measurements, are presented. For purely forced convection a 2D mean wind, which can be approximated by a solid body rotation, is found. With increasing Archimedes number this structure becomes unstable, leading to a transition of the solid body rotation into additional smaller convection rolls. Proper orthogonal decomposition of the instantaneous velocity fields has been performed for further analysis of these coherent large-scale structures. Their fingerprint is found in the spatial temperature distribution of the out flowing air at the end of the outlet channel, which reveals a temporally stable profile with two maxima over the length of the outlet. Moreover a maximum in the global enthalpy transport by the fluid is found at Ar ≈ 0.6.     

矩形悬臂薄板的解析解

首先把弹性薄板弯曲问题的控制方程表示成为Hamilton正则方程,然后利用辛几何方法对全状态相变量进行分离变量,求出其本征值后,再按本征函数展开的方法求出矩形悬臂薄板的解析解。由于在求解过程中不需要事先人为地选取挠度函数,而是从薄板弯曲的基本方程出发,直接利用数学的方法求出可以满足其边界条件的这类问题的解析解,使得问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文所采用的方法以及所推导出的公式的正确性。     

Magnetohydrodynamic flow in a rectangular duct

The magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct, with an external magnetic field applied transverse to the flow, has been investigated. One of the duct's boundaries which is perpendicular to the magnetic field is taken partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 100. All the infinite series obtained are transformed to infinite integrals first and then to finite integrals which contain modified Bessel functions of the second kind. In this way, the difficulties associated with the computation of infinite integrals with oscillating integrands and slowly converging infinite series, the convergence of which is further affected for large values of M, have been avoided. It is found that, as M increases, boundary layers are formed near the non-conducting boundaries and in the interface region, and a stagnant region is developed in front of the conducting boundary for velocity field. The maximm value of magnetic field takes place on the conducting part. These behaviours are shown on some graphs.     

Heat transfer and fluid dynamics of air–water two-phase flow in micro-channels

Heat transfer, pressure drop, and void fraction were simultaneously measured for upward heated air–water non-boiling two-phase flow in 0.51 mm ID tube to investigate thermo–hydro dynamic characteristics of two-phase flow in micro-channels. At low liquid superficial velocity j_l frictional pressure drop agreed with Mishima–Hibiki’s correlation, whereas agreed with Chisholm–Laird’s correlation at relatively high j_l. Void fraction was lower than the homogeneous model and conventional empirical correlations. To interpret the decrease of void fraction with decrease of tube diameter, a relation among the void fraction, pressure gradient and tube diameter was derived. Heat transfer coefficient fairly agreed with the data for 1.03 and 2.01 mm ID tubes when j_l was relatively high. But it became lower than that for larger diameter tubes when j_l was low. Analogy between heat transfer and frictional pressure drop was proved to hold roughly for the two-phase flow in micro-channel. But satisfactory relation was not obtained under the condition of low liquid superficial velocity.     

矩形槽道内表面活性剂减阻流体流场特性

采用粒子图像测速仪对矩形槽道内表面活性减阻流体在流动方向(x方向)与壁面垂直方向(y方向)所在平面的流场进行了测量,分析了速度、涡量、速度脉动相关量在流场内的瞬态分布,以及对500幅相同工况的流场进行了统计平均. 结果显示: 与牛顿流体相比, 表面活性剂减阻流体接近于层流流动,横向速度脉动被大幅减弱,导致湍流输运减弱,雷诺应力远远小于水. 减阻流体流向速度脉动呈条带特征,沿流动方向发展,反映了减阻流体不同于水的湍流输运特征.      

The second-law analysis of mixed convection in rectangular ducts

 The rate of entropy generation, S˙ ^′ _G[W/mK], is examined both theoretically and numerically for forced and mixed convection in a rectangular duct heated at the bottom. Under fully-developed flow conditions S˙ ^′ _G is expressed in terms of relevant non-dimensional hydrodynamic and thermal parameters. Numerically, it is demonstrated that S˙ ^′ _G is a single, effective parameter to examine both thermal and hydrodynamic fields and their variations. Received on 22 November 1999     

On pattern selection in mixed convection in rectangular ducts

Mixed convection in a horizontal rectangular duct has the same critical Rayleigh number as natural convection in a rectangular cavity for the onset of convection. The linear stability analysis predicts either an odd or an even number of convective rolls to appear depending on the aspect ratio of the cross section. However, it has been shown both experimentally and numerically that an even number of convective rolls appears under supercritical conditions for fully developed mixed convection. The paper first presents an analytical solution for the buoyancy-induced mainstream velocity, w ^b , at the onset of buoyancy-induced motion in a forced convective flow. Then, a comparison in the initial growth rate of w ^b is made between the case of an odd and an even number of rolls; which shows the selection of an even number of rolls over an odd number in mixed convection except for low aspect ratio ducts.     

Rectilinear oscillatory viscoelastic flow in rectangular ducts

Rectilinear oscillatory pressure-driven motion of Oldroyd fluids with complex dynamic viscosity within straight rectangular ducts is described. The linear viscoelastic flow solutions are given as functions of a Stokes number, the ratio of cross-section side lenghts and two Oldroyd time constants. Exact analysis and numerical approximate solutions via finite difference methods are shown, as well as suggestions for applying the theory to measure viscoelastic material properties.     

An experimental study of ice-layer transition phenomena in a rectangular duct containing water flow

Measurements of both the velocity and turbulence-intensity distributions above an ice-layer surface along flow direction have been performed to clarify the ice-layer transition phenomena observed in a rectangular duct. The test duct which has a lower cooled wall kept less than the freezing temperature of water with cross-sectional dimension of 50 mm by 19 mm was used in the present measurements. The velocity and turbulence-intensity distributions in the test duct were measured using Laser Doppler Velocimeter set up on the two-dimensional traversing table. The freezing experiments were carried out under the condition of uniform water-flow rate even after the ice layer has developed in the test duct. It was found that inlet water flow tended to be laminarized under an influence of developing ice layer, and that onset of the ice-layer transition phenomena might be closely related to an increase in turbulence intensity in the water flow above the developing ice-layer surface.