Interaction between a cantilevered-free flexible plate and ideal flow

We develop a new computational model of the linear fluid–structure interaction of a cantilevered flexible plate with an ideal flow in a channel. The system equation is solved via numerical simulations that capture transients and allow the spatial variation of the flow–structure interaction on the plate to be studied in detail. Alternatively, but neglecting wake effects, we are able to extract directly the system eigenvalues to make global predictions of the system behaviour in the infinite-time limit. We use these complementary approaches to conduct a detailed study of the fluid–structure system. When the channel walls are effectively absent, predictions of the critical velocity show good agreement with those of other published work. We elucidate the single-mode flutter mechanism that dominates the response of short plates and show that the principal region of irreversible energy transfer from fluid to structure occurs over the middle portion of the plate. A different mechanism, modal-coalescence flutter, is shown to cause the destabilisation of long plates with its energy transfer occurring closer to the trailing edge of the plate. This mechanism is shown to allow a continuous change to higher-order modes of instability as the plate length is increased. We then show how the system response is modified by the inclusion of channel walls placed symmetrically above and below the flexible plate, the effect of unsteady vorticity shed at the trailing edge of the plate, and the effect of a rigid surface placed upstream of the flexible plate. Finally, we apply the modelling techniques in a brief study of upper-airway dynamics wherein soft-palate flutter is considered to be the source of snoring noises. In doing so, we show how a time-varying mean flow influences the type of instability observed as flow speed is increased and demonstrate how localised stiffening can be used to control instability of the flexible plate.     

Viscous incompressible flow in a narrow channel with one free wall and fluid supply through a porous insert

The problem investigated relates the plane unsteady flow of a viscous incompressible fluid in a narrow channel one of whose walls is free and acted upon by a given load, while the other is rigidly fixed. The fluid enters the channel through a porous insert in the stationary wall. A model of the flow of a thin film of viscous incompressible fluid and Darcy's law for flow in a porous medium are used to find the distribution of fluid pressure and velocity in the channel and the porous insert in the two-dimensional formulation for fairly general boundary conditions in the case where the length of the porous insert exceeds the length of the free wall. In the particular case where the length of the porous insert is equal to the length of the free wall an exact stationary solution of the problem is obtained for a given value of the channel height. The stability of the equilibrium position of the free wall supported on a hydrodynamic fluid film is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–24, January–February, 1986.     

Hydroelastic analysis of gravity wave interaction with submerged horizontal flexible porous plate

The present study deals with the surface gravity wave interaction with submerged horizontal flexible porous plate under the assumption of small amplitude water wave theory and structural response. The flexible porous plate is modeled using the thin plate theory and wave past porous structure is based on the generalized porous wavemaker theory. The wave characteristics due to the interaction of gravity waves with submerged flexible porous structure are studied by analyzing the complex dispersion relation using contour plots. Three different problems such as (i) wave scattering by a submerged flexible porous plate, (ii) wave trapping by submerged flexible porous plate placed at a finite distance from a rigid wall and (iii) wave reflection by a rigid wall in the presence of a submerged flexible porous plate are analyzed. The role of flexible porous plate in attenuating wave height and creating a tranquility zone is studied by analyzing the reflection, transmission and dissipation coefficients for various wave and structural parameters such as angle of incidence, depth of submergence, plate length, compression force and structural flexibility. In the case of wave trapping, the optimum distance between the porous plate and rigid wall for wave reflection is analyzed in different cases. In addition, effects of various physical parameters on free surface elevation, plate deflection, wave load on the plate and rigid wall are studied. The present approach can be extended to deal with acoustic wave interaction with flexible porous plates.     

Free vibration of an elastic bottom plate of a partially fluid-filled cylindrical container with an internal body

An analytical method is developed to consider the free vibration of an elastic bottom plate of a partially fluid-filled cylindrical rigid container with an internal body. The internal body is a rigid cylindrical block that is concentrically and partially submerged inside the container. The developed method captured the analytical features of the velocity potential in a non-convex, continuous, and simply connected fluid domain including the interaction between the fluid and the structure. The interaction between the fluid and the bottom plate is included. The Galerkin method is used for matching the velocity potentials appropriate to two distinct fluid regions across the common horizontal boundary (artificial horizontal boundary). Then, the Rayleigh–Ritz method is also used to calculate the natural frequencies and modes of the bottom plate of the container. The results obtained for the problem without internal body are in close agreement with both experimental and numerical results available in the articles. A finite element analysis is also used to check the validity of the present method in the presence of the internal body. Furthermore, the influences of various variables such as fluid level, internal body radius, internal body length, and the number of nodal diameters and circles on the dynamic behaviour of the coupled system are investigated.     

The flexibility effect of a plate according to various angles of attack in a free-stream

This paper presents a computational fluid–structure interaction analysis for a flexible plate in a free-stream to investigate the effects of flexibility and angle of attack on force generation. A Lattice Boltzmann Method with an immersed boundary technique using a direct forcing scheme model of the fluid is coupled to a finite element model with rectangular bending elements. We investigated the effects of various angles of attack of a flexible plate fixed at one of the end edges in a free-stream at a Reynolds number of 5000, which represents the wing flapping condition of insects and small birds in nature. The lift of the flexible plate is maintained at the large angle of attack, whereas the rigid plate shows the largest lift at angles of attack around 30–40° and then drastic reductions in the lift at the large angle of attack. If we consider the efficiency as the lift divided by the drag, the flexible plate shows better efficiency at angles of attack greater than 30° compared to the rigid plate. The better performance of the flexible plate at large angles of attack comes from the deformation of the plate, which produces an interaction between the trailing edge vortex and the short edge vortex. The horseshoe-shaped vortex produced by a large vortex interaction at the trailing edge side has an important role in increasing the lift, and the small projection area due to the deformation reduces the drag. Furthermore, we investigate the role of flexibility on the lift and the drag force of the rectangular plate in a free-stream as the Reynolds number increases. Whenever a large vortex interaction at the trailing edge side is shown, the efficiency of the rectangular plate is improved. Especially, the flexible plate shows better efficiency as the Reynolds number increases regardless of the angle of attack.     

Numerical and asymptotic study of the two-dimensional problem of the hydroelastic behavior of a floating plate in waves

The problem of the behavior of a floating elastic plate in waves is solved numerically. The normal mode method is used. For a fluid of finite depth, the hydrodynamic coefficients are obtained in explicit form. Numerical results are compared with experimental data for the stress distribution in the plate and also with numerical results of other authors. The results are in good agreement for not very short waves. For incident waves whose wavelength is comparable with the length of the plate, a long-wave approximation of the solution is proposed. Within the framework of this approximation, the solution is given in analytical form. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 90–96, March–April, 2000.     

受限亚音速气流中倒置悬臂壁板静气弹稳定性的理论及实验研究

板壳结构在航空航天、高速列车、能量采集等诸多工程领域已经得到了广泛应用. 将悬臂壁板倒置于轴向气流中并在壁板周围流场中设置刚性壁面可有效地调控壁板的失稳速度, 是俘能器优化设计的重要措施之一. 但针对刚性壁面作用下亚音速气流中倒置悬臂壁板的失稳机制仍需要开展深入研究. 本文以受限亚音速气流中倒置的二维悬臂壁板为对象, 以理论分析及风洞实验为手段, 研究了单侧刚性壁面效应对倒置悬臂壁板静态失稳特性的影响规律. 在理论分析中, 首先应用镜像函数法来处理壁面约束条件, 基于算子理论研究获得了以Possio积分方程为表征的壁板气动力, 壁面效应实际表征为一包含移位Tricomi算子的复合算子; 然后将壁板失稳方程的求解问题转化为定区间上的函数逼近问题; 最后, 依据Wererstrass定理并利用最小二乘法求解该最优函数, 以获得系统的失稳临界参数. 在试验研究中依据压杆稳定原理设计了壁板静态失稳的测试方法并完成了风洞实验. 理论分析结果表明, 壁板会发生发散(静气动弹性)失稳, 临界动压随壁板与壁面间距的增加而增大并最终趋于稳定(无壁面情况); 通过理论与风洞实验结果的对比分析, 验证了本文气动力及理论分析的适用性及准确性. 针对倒置悬臂壁板结构的气动弹性失稳问题, 本文提出的方法不涉及系统方程的离散及特征值求解问题, 而是将其转化为了定区间上的函数逼近问题进行求解, 这为弹性结构静气动弹性失稳问题的研究提供了一个可行的新思路.      

Self-sustained oscillations of a confined impinging jet

The present paper investigates the dynamics of a laminar plane jet impinging on a flat plate in a channel. An experimental parametric study is carried out to determine the flow regimes at different levels of confinement and Reynolds numbers. For very confined jets, the flow is steady whatever the Reynolds number. The overall structure of the flow is symmetric with respect to the jet axis and is characterized by the presence of recirculation zones at the channel walls. The dynamics is radically different for less confined jets. Above a critical Reynolds number, the flow bifurcates in the form of an oscillating flapping mode of the impinging jet. Analyses of the experimental results provide with a quantitative characterization of this regime in terms of amplitude, wavelength and frequency. This self-oscillating bifurcated flow induces strong sweepings of the target plate by the jet and intense vortex dipole ejections from the impacted wall. Such a regime is expected to be particularly useful in the enhancement of the local heat transfer at relatively low cost in terms of flow rate.     

Plane-Couette flow between smooth and rough walls

A novel approach is utilized to acquire new experimental information on plane-Couette flow. To this end a rigid plate, functioning as the moving wall, is propelled through air by the carriage of a towing channel, and the fixed wall is a stationary bench. It is shown that, irrespective of the state of the walls, the critical Reynolds number, expressed in terms of relative wall velocity and wall spacing, is about 1,200. Findings support the notion of universality of turbulent flow structure in the wall region, but shed doubt on the conjecture of homogeneous shear-flow turbulence in the core.     

Flow of two stably stratified fluids in an open channel with addition of fluids along the channel length

It is shown that two stably stratified fluids flowing in an open channel have two critical flow conditions. The one at higher flowrates is equivalent to the choked flow condition of a single fluid over a broad-crested weir, when the Froude number is unity. The lower critical condition imposes restrictions, which define the system if fluids are added progressively along the channel length and the flowrates increase from low to high values. However, if the flowrate does not become sufficiently large to pass through the lower critical condition, this condition will then define a form of choking, which again determines the system.It is shown that an important special case, with the proportional flowrates of the two fluids kept constant, has an analytical solution in which the relative depths of the fluids is a constant along the channel. Other systems must be solved numerically.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

Finite Oscillations of a Cylinder in a Coaxial Duct Subjected to Annular Compressible Flow

As a generalization considering small fluid-structural vibrations, the present paper examines the finite magnitude oscillatory motion of an elastically supported rigid cylinder in a cylindrical rigid duct conveying a compressible flow. The fluid is assumed to be inviscid and irrotational and free purely transverse vibrations of the body are dealt with. The governing equations of motion are the fully nonlinear Euler equations together with the continuity equation and a state equation (here for an ideal gas), the ordinary differential equation for the vibrating cylinder, and the kinematical transition and boundary conditions at the moving contact interface between fluid and body and the outside fluid border, respectively. A pertubation analysis is performed to calculate not only the dynamic characteristics for small coupled oscillations but also the corrections due to the inherent nonlinearities of the vibroacoustic problem. To make the calculation steps more transparent, the simpler problem of a two-dimensional channel flow between a rigid wall and an elastically supported rigid plate is also included in the present study. As an outlook, the influence of flexibility of the cylinder (or the plate) is addressed and the problem of forced vibrations is touched. This revised version was published online in July 2006 with corrections to the Cover Date.     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

Effects of turbulence and secondary flows on subcooled flow boiling

Experiments are conducted on the influence of turbulence and longitudinal vortices on subcooled flow boiling in a vertical, rectangular channel. Different flow inserts are used to create turbulence and vortices in the channel. Studied boiling regimes range from the onset of nucleate boiling over the critical heat flux up to fully developed film boiling. A wide range of measuring techniques is applied: time averaged particle image velocimetry (PIV) is used in cold flows for the evaluation of the effects the inserts have on the flow, high speed PIV and photography are used to determine the effects on the fluid and vapor movement in boiling experiments. Digital Holographic Interferometry is used for the evaluation of temperature distributions in the boiling flow. Furthermore, optical microprobes are used to obtain pointwise measurements in areas inaccessible to the imaging techniques. The experiments show that the flow inserts can have considerable impact on the heat fluxes and the distribution of vapor and temperature along the channel. All used inserts lead to an increase in critical heat flux, which is more pronounced for stronger turbulence and higher flow rates and fluid subcoolings. The measuring techniques reveal both a better transport of vapor from the heater surface as well as an increase in mixing in the liquid phase with flow inserts.     

复合柔性结构全局模态函数提取与状态空间模型构建

随着航空航天等领域中实际工程结构的大型化和柔性化,结构的非线性振动和主动振动控制问题越来越凸显.分析和处理此类结构出现的复杂振动问题的关键在于建立系统的非线性动力学模型与状态空间模型.对于由柔性部件、刚体、连接部件构成的复合柔性结构,由于各部件之间的振动耦合效应,单个柔性部件在悬臂、简支和自由等静定边界下的模态与结构的真实模态有较大差异.为此,本文提出复合柔性结构全局模态的解析提取方法,通过全局模态离散得到系统非线性动力学模型,从而构建状态空间模型.该方法采用笛卡尔坐标描述系统的运动,建立系统的运动方程;结合描述柔性部件的偏微分方程、刚体的常微分运动方程、连接界面处力、力矩、位移和转角的匹配条件以及系统的边界条件,利用分离变量法给出统一形式的频率方程,获取系统的固有频率和解析函数表征的全局模态.这里提出的全局模态提取方法不仅便于复合柔性结构固有频率和全局模态的参数化分析,而且为建立复合柔性结构低维非线性动力学模型和状态空间模型提供了有效的途径,对于推进这类结构的非线性动力学分析与主动振动控制研究具有重要意义.      

Finite Difference Method for the Acoustic Radiation of an Elastic Plate Excited by a Turbulent Boundary Layer: A Spectral Domain Solution

A finite difference method is developed to study, on a two-dimensional model, the acoustic pressure radiated when a thin elastic plate, clamped at its boundaries, is excited by a turbulent boundary layer. Consider a homogeneous thin elastic plate clamped at its boundaries and extended to infinity by a plane, perfectly rigid, baffle. This plate closes a rectangular cavity. Both the cavity and the outside domain contain a perfect fluid. The fluid in the cavity is at rest. The fluid in the outside domain moves in the direction parallel to the system plate/baffle with a constant speed. A turbulent boundary layer develops at the interface baffle/plate. The wall pressure fluctuations in this boundary layer generates a vibration of the plate and an acoustic radiation in the two fluid domains. Modeling the wall pressure fluctuations spectrum in a turbulent boundary layer developed over a vibrating surface is a very complex and unresolved task. Ducan and Sirkis [1] proposed a model for the two-way interactions between a membrane and a turbulent flow of fluid. The excitation of the membrane is modeled by a potential flow randomly perturbed. This potential flow is modified by the displacement of the membrane. Howe [2] proposed a model for the turbulent wall pressure fluctuations power spectrum over an elastomeric material. The model presented in this article is based on a hypothesis of one-way interaction between the flow and the structure: the flow generates wall pressure fluctuations which are at the origin of the vibration of the plate, but the vibration of the plate does not modify the characteristics of the flow. A finite difference scheme that incorporates the vibration of the plate and the acoustic pressure inside the fluid cavity has been developed and coupled with a boundary element method that ensures the outside domain coupling. In this paper, we focus on the resolution of the coupled vibration/interior acoustic problem. We compare the results obtained with three numerical methods: (a) a finite difference representation for both the plate displacement and the acoustic pressure inside the cavity; (b) a coupled method involving a finite difference representation for the displacement of the plate and a boundary element method for the interior acoustic pressure; (c) a boundary element method for both the vibration of the plate and the interior acoustic pressure. A comparison of the numerical results obtained with two models of turbulent wall pressure fluctuations spectrums - the Corcos model [3] and the Chase model [4] - is proposed. A difference of 20 dB is found in the vibro-acoustic response of the structure. In [3], this difference is explained by calculating a wavenumber transfer function of the plate. In [6], coupled beam-cavity modes for similar geometry are calculated by the finite difference method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Coupled free vibration analysis of a fluid-filled rectangular container with a sagged bottom membrane

In the present paper, two-dimensional coupled free vibrations of a fluid-filled rectangular container with a sagged bottom membrane are investigated. This system consists of two rigid walls and a membrane anchored along two rigid vertical walls. It is filled with incompressible and inviscid fluid. The membrane material is assumed to act like an inextensible material with no bending resistance. First, the nonlinear equilibrium equation is solved and the equilibrium shape of the membrane is obtained using an analytical formulation neglecting the membrane weight. The small vibrations about the equilibrium configuration are then investigated. Along the contact surface between the bottom membrane and the fluid, the compatibility requirement is applied for the fluid–structure interactions and the finite element method is used to calculate the natural frequencies and mode shapes of the fluid–membrane system. The vibration analysis of the coupled system is accomplished by using the displacement finite element for the membrane and the pressure fluid-finite element for the fluid domain. The variations of natural frequencies with the pressure head, the membrane length, the membrane weight and the distance between two rigid walls are examined. Moreover, the mode shapes of system are investigated.     

Interaction between a simplified soft palate and compressible viscous flow

Fluid–structure interaction in a simplified 2D model of the upper airways is simulated to study flow-induced oscillation of the soft palate in the pharynx. The goal of our research has been a better understanding of the mechanisms of the Obstructive Sleep Apnea Syndrome and snoring by taking into account compressible viscous flow. The inspiratory airflow is described by the 2D compressible Navier–Stokes equations, and the soft palate is modeled as a flexible plate by the linearized Euler–Bernoulli thin beam theory. Fluid–structure interaction is handled by the arbitrary Lagrangian–Eulerian formulation. The fluid flow is computed by utilizing 4th order accurate summation by parts difference operators and the 4th order accurate classical Runge–Kutta method which lead to very accurate simulation results. The motion of the cantilevered plate is solved numerically by employing the Newmark time integration method. The numerical schemes for the structure are verified by comparing the computed frequencies of plate oscillation with the associated second mode eigenfrequency in vacuum. Vortex dynamics is assessed for the coupled fluid–structure system when both airways are open and when one airway is closed. The effect of mass ratio, rigidity and damping coefficient of the plate on the oscillatory behavior is investigated. An acoustic analysis is carried out to characterize the acoustic wave propagation induced by the plate oscillation. It is observed that the acoustic wave corresponding to the quarter wave mode along the length of the duct is the dominant frequency. However, the frequency of the plate oscillation is recognizable in the acoustic pressure when reducing the amplitude of the quarter wave mode.     

An efficient method for vibration and stability analysis of rectangular plates axially moving in fluid

An efficient method is developed to investigate the vibration and stability of moving plates immersed in fluid by applying the Kirchhoff plate theory and finite element method. The fluid is considered as an ideal fluid and is described with Bernoulli’s equation and the linear potential flow theory. Hamilton’s principle is used to acquire the dynamic equations of the immersed moving plate. The mass matrix, stiffness matrix,and gyroscopic inertia matrix are determined by the exact analytical integ...     

Asymmetric vibrations and stability of a rotating annular plate loaded by a torque

The paper investigates transverse vibration of a thin annular plate clamped at its inner edge to a rigid shaft, while its outer edge is clamped to a rigid cylinder. The shaft and the outer edge of the plate are loaded by torques of the same intensity, but of opposite directions. The whole structure rotates at a constant angular speed. The solution has been determined using Galerkin’s method. The obtained results illustrate the impact of the torque, angular speed and inner and outer radia ratio to transverse asymmetric vibration frequency of the plate. Stability of the plate has been examined and critical values of angular speed and torque leading to the loss of stability of the plate have been determined. Some mode shapes have been drawn and the influence of torque and angular speed on nodal lines has been shown.