Hydroelastic stability of a rectangular plate interacting with a layer of ideal flowing fluid

The three-dimensional formulation of the problem on the natural vibrations and stability of an elastic plate which interacts with a quiescent or flowing fluid is represented and a finite element algorithm of its numerical implementation is proposed. The governing equations, which describe vortex-free ideal fluid dynamics in the case of small perturbations, are written in terms of the perturbation velocity potential and transformed using the Bubnov–Galerkin method. The plate strains are determined on the basis of the Timoshenko theory. The variational principle of virtual displacements which takes into account the work done by inertial forces and the hydrodynamic pressure is used for the mathematical formulation of the dynamic problem of elastic structure. The solution of the problem is reduced to calculations and an analysis of complex eigenvalues of a coupled system of two equations. The effect of the fluid layer height on the eigenfrequencies and the critical velocities of the loss of stability is estimated numerically. It is shown that there exist different types of instability determined by combinations of the kinematic boundary conditions prescribed at the plate edges.     

应用有限差分法的轴向流作用下叠层板的稳定性和模态分析

分析了轴向流作用下两端简支和固支叠层板的稳定性。基于势流理论建立轴向流作用下叠层板的流固耦合系统连续型运动方程,基于有限差分法建立了流场网格和结构网格统一的离散化格式,流场势函数用板的横向振动位移变量来表示,得到关于叠层板的横向振动位移变量的控制方程。求解控制方程的广义特征值,计算分析结果表明,两端简支和两端固支模型发生屈曲失稳,且得到了屈曲失稳临界速度与叠层板的层数和无量纲板间距的关系。此外,轴向流作用下叠层板的一阶模态并不是叠层板的同相弯曲模态。     

Stability Analysis of Maxwell Viscoelastic Pipes Conveying Fluid with Both Ends Simply Supported

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Small scale effect on flow-induced instability of double-walled carbon nanotubes

Small scale effect on flow-induced instability of double-walled carbon nanotubes (DWCNTs) is investigated using an elastic shell model based on Donnell’s shell theory. The dynamic governing equations of DWCNTs are formulated on the basis of nonlocal elasticity theory, in addition, the van der Waals (vdW) interaction between the inner and outer walls is taken into account in the nonlocal shell modeling. The instability of DWCNTs that is induced by a pressure-driven steady flow is investigated. The numerical computations indicate that as the flow velocity increases, DWCNTs have a destabilizing way to get through multi-bifurcations of the first and second bifurcations in turn. It is concluded that the natural frequency of DWCNTs and the critical flow velocity of the flow-induced instability are strictly related to the ratio of the length to the outer radius of DWCNTs, the pressure of the fluid and the small scale effects. Furthermore, it is interesting to observe that as the small scale effects are considered, the natural frequencies and the critical flow velocities of DWCNTs decrease as compared to the results with the classical (local) continuum mechanics, therefore, the small scale effects play an important role on performing the instability analysis in the fluid-conveying DWCNTs.     

Critical velocities in fluid-conveying single-walled carbon nanotubes embedded in an elastic foundation

The problem of stability of fluid-conveying carbon nanotubes embedded in an elastic medium is investigated in this paper. A nonlocal continuum mechanics formulation, which takes the small length scale effects into consideration, is utilized to derive the governing fourth-order partial differential equations. The Fourier series method is used for the case of the pinned–pinned boundary condition of the tube. The Galerkin technique is utilized to find a solution of the governing equation for the case of the clamped–clamped boundary. Closed-form expressions for the critical flow velocity are obtained for different values of the Winkler and Pasternak foundation stiffness parameters. Moreover, new and interesting results are also reported for varying values of the nonlocal length parameter. It is observed that the nonlocal length parameter along with the Winkler and Pasternak foundation stiffness parameters exert considerable effects on the critical velocities of the fluid flow in nanotubes.     

温度场作用下输流悬臂碳纳米管的颤振失稳分析

以非局部弹性理论为基础,采用欧拉-伯努利梁模型,考虑碳纳米管的小尺度效应,应用哈密顿原理获得了温度场作用下的输流悬臂单层碳纳米管（SWCNT）的振动控制方程以及边界条件,依靠微分变换法（DTM法）对此高阶偏微分方程进行求解,通过数值计算研究了温度场中悬臂单层输流碳纳米管的振动与颤振失稳问题。结果表明：管内流体流速、温度场中温度变化情况与小尺度参数都会对系统振动频率以及颤振失稳临界流速产生影响。其中,小尺度效应将会降低悬臂输流系统的稳定性,使系统更为柔软;而高温场与低温场对系统动态失稳的影响不同,低温场中随温度变化值的增加,系统的稳定性提高;高温场这一作用效果恰好与之相反。     

EFFECT OF ELASTIC FOUNDATIONS ON DIVERGENCE AND FLUTTER OF AN ARTICULATED PIPE CONVEYING FLUID

In this study, we generalize earlier investigations of Benjamin and Sugiyama & Paı̈doussis devoted to the stability of articulated pipes conveying fluid. The present study additionally incorporates the translational and rotational elastic foundations in an attempt to answer the following question: Do the elastic foundations increase the critical velocity of the fluid? It turns out that the attachment of the elastic foundation along the entire length of the pipe may either strengthen or weaken the system, with attendant increase or decrease in the critical velocity. The physical mechanism of the change of type of instability plays a crucial role in deciding whether or not the elastic foundation increases the critical velocity. If the elastic foundations are attached within the first pipe only, the instability mechanism is by flutter. If the elastic foundations are attached beyond the first pipe, then divergence may occur. The interplay of the two mechanisms may lead to a decrease of the critical velocity of the system with elastic foundations. A remarkable nonmonotonous dependence of the critical velocity with respect to the attachment foundation ratio is established.     

Effect of small parameters on the structure of the front formed by unstable viscous-fluid displacement from a Hele-Shaw cell

The process of displacement of a viscous fluid from a Hele-Shaw cell consisting of two plates separated by a small gap is investigated. The front formed when the fluid is displaced from the cell by another, lower-viscosity fluid is unstable. The lower-viscosity fluid breaks through the layer of displaced fluid and forms channels called viscous fingers. As a result, a mixing zone occupied by both displaced and displacing fluids is formed. The structure of the unstable displacement front is investigated when the surface tension forces have no effect on the shape of the fingers. This situation is realized when a water-glycerin mixture is rapidly displaced from the cell by water. Equations taking the inertial and viscous forces acting in the plane of the plates into account are obtained by averaging the Navier-Stokes equations over the cell gap. Using the equations obtained the stability of a plane displacement front traveling in the direction of its normal and the stability of the lateral surfaces of the viscous fingers is investigated when the fluid velocities are parallel to the interface. From the solution for stability of the transverse displacement front it follows that the viscous forces acting in the plane of the plates determine the finger width (when there is no surface tension). Instability also develops in the flow on the longitudinal fluid interface. In this case the destabilizing factor is the inertial forces. Under the action of this instability the fingers, in their turn, lose stability and disintegrate into viscous bubbles.     

THE CROSSING FREQUENCY AS A MEASURE OF HEAT EXCHANGER SUPPORT-PLATE EFFECTIVENESS

The crossing frequency is the number of times per second the vibration amplitude crosses the zero displacement line from negative displacement to positive displacement. In flow-induced vibration in which the motions are often random and/or a number of modes contribute to the vibration amplitudes, the crossing frequencies are modal-weighted average frequencies of the vibration. It is postulated in this paper that the crossing frequency can be used as a measure of heat exchanger support-plate effectiveness. Using a time-domain, nonlinear analysis technique, the crossing frequencies of a tube vibrating in support plates with oversized holes can be computed as a function of time and the tube-to-support-plate clearances. It was found that the fluid–elastic stability margin of a tube bundle, in the context of the original Connors' equation for tube bundle fluid–elastic instability, should be independent of the tube-to-support-plate clearances. A simple method of estimating the critical velocity based on the time-domain equation of fluid–elastic stability is suggested.     

Pressure dependence of the instability of multiwalled carbon nanotubes conveying fluids

Based on an elastic beam model, the instability of multiwalled carbon nanotubes (MWCNTs) induced by the moving fluid inside is investigated. At critical flow velocities, the MWCNTs become unstable and undergo pitchfork bifurcation and subsequently Hopf bifurcation. These critical velocities are found to increase very quickly with respect to decreasing inner radius and are inversely proportional to the length-to-outer-radius ratio. The effect of the van der Waals (vdW) interaction between tubes is investigated and it is found that the vdW interaction can enhance the stability of MWCNTs in general, but the vdW interaction reduces the stability capacity of MWCNTs with very small inner radius.     

Stability of a flexible insert in one wall of an inviscid channel flow

A hybrid of computational and theoretical methods is extended and used to investigate the instabilities of a flexible surface inserted into one wall of an otherwise rigid channel conveying an inviscid flow. The computational aspects of the modelling combine finite-difference and boundary-element methods for structural and fluid elements respectively. The resulting equations are coupled in state-space form to yield an eigenvalue problem for the fluid–structure system. In tandem, the governing equations are solved to yield an analytical solution applicable to inserts of infinite length as an approximation for modes of deformation that are very much shorter than the overall length of the insert. A comprehensive investigation of different types of inserts – elastic plate, damped flexible plate, tensioned membrane and spring-backed flexible plate – is conducted and the effect of the proximity of the upper channel wall on stability characteristics is quantified. Results show that the presence of the upper-channel wall does not significantly modify the solution morphology that characterises the corresponding open-flow configuration, i.e. in the absence of the rigid upper channel wall. However, decreasing the channel height is shown to have a very significant effect on instability-onset flow speeds and flutter frequencies, both of which are reduced. The channel height above which channel-confinement effects are negligible is shown to be of the order of the wavelength of the critical mode at instability onset. For spring-backed flexible plates the wavelength of the critical mode is much shorter than the insert length and we show very good agreement between the predictions of the analytical and the state-space solutions developed in this paper. The small discrepancies that do exist are shown to be caused by an amplitude modulation of the critical mode on an insert of finite length that is unaccounted for in the travelling-wave assumption of the analytical model. Overall, the key contribution of this paper is the quantification of the stability bounds of a fundamental fluid–structure interaction (FSI) system which has hitherto remained largely unexplored.     

Dynamic problems for and stress–strain state of inhomogeneous shell structures under stationary and nonstationary loads

This paper reviews studies and analyzes results on the effect of discrete ribs on the dynamic characteristics of rectangular plates and cylindrical shells. Use is made of the vibration equations derived from the classical theories of beams, plates, and shells. The effect of Pasternak’s elastic foundation on the critical velocities of a structurally orthotropic model of a ribbed cylindrical shell is determined. Nonstationary problems are solved for perforated and ribbed shells of revolution filled with a fluid or resting on an elastic foundation and subjected to moving or impulsive loads. Results from studies of the behavior of sandwich shell structures under impulsive loads of various types are presented     

On the instability of an axially moving elastic plate

Problems of stability of an axially moving elastic band travelling at constant velocity between two supports and experiencing small transverse vibrations are considered in a 2D formulation. The model of a thin elastic plate subjected to bending and tension is used to describe the bending moment and the distribution of membrane forces. The stability of the plate is investigated with the help of an analytical approach. In the frame of a general dynamic analysis, it is shown that the onset of instability takes place in the form of divergence (buckling). Then the static forms of instability are investigated, and critical regimes are studied as functions of geometric and mechanical problem parameters. It is shown that in the limit of a narrow strip, the 2D formulation reduces to the classical 1D model. In the limit of a wide band, there is a small but finite discrepancy between the results given by the 1D model and the full 2D formulation, where the discrepancy depends on the Poisson ratio of the material. Finally, the results are illustrated via numerical examples, and it is observed that the transverse displacement becomes localised in the vicinity of free boundaries.     

Experimental investigation of dynamic stability of a cantilever pipe aspirating fluid

The dynamic stability of a submerged cantilever pipe conveying fluid from the free end to the fixed one is considered as one of the unresolved issues in the area of fluid–structure interaction. There is a contradiction between theoretical predictions and experiments. Reported experiments did not show any instability, while theory predicts instability beyond a critical fluid velocity. Recently, several papers appeared, improving the theoretical modelling of pipe dynamics. All theories predict instability, either oscillatory or static, referred to here as flutter and divergence, respectively. A new test set-up was designed to investigate the hypothesis that previous experimental set-ups could not allow observations of pipe instability or the pipe aspirating water is unconditionally stable. In this new test set-up, the fluid velocity could exceed the theoretically predicted critical velocities. A cantilever pipe of about 5 m length was partly submerged in water. The free open end of the pipe was in the water, whereas the fixed end was above the waterline. The experiments clearly showed that the cantilever pipe aspirating water is unstable beyond a critical velocity of water convection through the pipe. Below this velocity the pipe is stable, whereas above it the pipe shows a complex motion that consists of two alternating phases. The first phase is a nearly periodic orbital motion with maximum amplitude of a few pipe diameters, whereas the second one is a noise-like vibration with very small amplitudes. Increasing the internal fluid velocity results in a larger amplitude of the orbital motion, but does not change the pipe motion qualitatively.     

Heat transfer effects on the stability of low speed plane Couette-Poiseuille flow

The stability problem of low-speed plane Couette-Poiseuille flow of air under heat transfer effects is solved numerically using the linear stability theory. Stability equations obtained from two-dimensional equations of motion and their boundary conditions result in an eigenvalue problem that is solved using an efficient shoot-search technique. Variable fluid properties are accounted for both in the basic flow and the perturbation (stability) equations. A parametric study is performed in order to assess the roles of moving wall velocity and heat transfer. It is found that the moving wall velocity and the location of the critical layers play decisive roles in the instability mechanism. The flow becomes unconditionally stable whenever the moving wall velocity exceeds half of the maximum velocity in the channel. With wall heating and Mach number effects included, the flow is stabilized.     

Stability of fluid-conveying periodic shells on an elastic foundation with external loads

This paper presents the beam-mode stability of a fluid-conveying periodic shell on an elastic foundation subjected to external loading. A transfer matrix (TM) method was developed to investigate the characteristics of steady-state waves in the system and the dynamic response of the periodic shell system. When subjected to external perturbations, including either a moving load or a stationary one, the shell may be subjected to instability for flow velocities exceeding a certain critical velocity. The system can also become unstable when a travelling load exceeds a certain critical value. The coupled effects of the speed of a moving load and the flow velocity of a fluid on the stability of the shell system were also investigated. A periodic structure was designed for such a shell system to enhance its dynamic stability. The periodic shell system produces innumerable velocity band gaps (VBGs), which could raise the critical velocity and extend the stable range for both the moving load and the flowing fluid. Finally, the formation mechanism of the VBGs was studied, as well as the effects of the thickness, length of the shell cells, Young׳s modulus and stiffness of the elastic foundation on modulating the VBGs.     

Exact solutions of incompressible Couette flow with porous walls for slightly rarefied gases

In this work, the transient incompressible Couette flow and steady-state temperature profiles between two porous parallel plates for slightly rarefied gases are solved exactly. The first-order approximation of slip velocity at the boundaries is used in the formulation. The solution is also applicable for Couette flow in micro-channels under certain circumstances. The influences of mass transfer and a nondimensional slip parameter on slip velocities are discussed. It is also found that the transient slip velocities at the walls are greatly different from the steady-state velocity slips. The influences of velocity slip and temperature slip parameters on the temperature distribution and heat transfer at the walls are analyzed and discussed. It is shown that the slip parameters can greatly change the temperature profiles and heat transfer characteristics at the walls.     

Linear stability of a two-dimensional uniform fluidized bed

Basic equations in a two-dimensional fluidized bed are constructed for the particle and the fluid phases, and linear stability to two-dimensional disturbances of the volume fractions and the velocities of both phases is analyzed. The diffusion of particles and an effective viscosity in the particle phase are considered. It was found that the inertia term due to the average fluid velocity is responsible for the instability, while the particle diffusion and the effective particle viscosity suppress the growth of disturbances. It was also found that the most unstable state has a vertical wavenumber vector.     

The Absolute Instability of Thin Wakes in an Incompressible/Compressible Fluid

RID="ID=" Communicated by P. HallAbstract:The absolute/convective instability of two-dimensional wakes forming behind a flat plate and near the trailing-edge of a thin wedge-shaped aerofoil in an incompressible/compressible fluid is investigated. The mean velocity profiles are obtained by solving numerically the classical compressible boundary-layer equations with a negative pressure gradient for the flat plate case, and the incompressible triple-deck equations for a thin wedge-shaped trailing-edge. In addition for a Joukowski aerofoil the incompressible mean boundary-layer flow in the vicinity of the trailing-edge is also calculated by solving the interactive boundary-layer equations. A linear stability analysis of the boundary-layer profiles shows that a pocket of absolute instability occurs downstream of the trailing-edge with the extent of the instability region increasing with more adverse pressure gradients. The region of absolute instability persists along the near-wake axis, while the majority of the wake is convectively unstable. For a thin wedge-shaped trailing-edge in an incompressible fluid, a similar stability analysis of the velocity profiles obtained via a composite expansion, also shows the occurrence of absolute instability behind the trailing-edge for a wedge angle greater than a critical value. For increasing values of the wedge angle and for thicker aerofoils, separation takes place near the trailing-edge and the extent of absolute instability increases. Calculations also show that for insulated plates compressibility has a stabilizing effect but cooling the wall destabilizes the flow unlike wall heating.} Received 11 May 1998 and accepted 25 February 1999     

Modeling of the phase lag causing fluidelastic instability in a parallel triangular tube array

Fluidelastic instability is considered a critical flow induced vibration mechanism in tube and shell heat exchangers. It is believed that a finite time lag between tube vibration and fluid response is essential to predict the phenomenon. However, the physical nature of this time lag is not fully understood. This paper presents a fundamental study of this time delay using a parallel triangular tube array with a pitch ratio of 1.54. A computational fluid dynamics (CFD) model was developed and validated experimentally in an attempt to investigate the interaction between tube vibrations and flow perturbations at lower reduced velocities U_r=1–6 and Reynolds numbers Re=2000–12 000. The numerical predictions of the phase lag are in reasonable agreement with the experimental measurements for the range of reduced velocities U_g/fd=6–7. It was found that there are two propagation mechanisms; the first is associated with the acoustic wave propagation at low reduced velocities, U_r<2, and the second mechanism for higher reduced velocities is associated with the vorticity shedding and convection. An empirical model of the two mechanisms is developed and the phase lag predictions are in reasonable agreement with the experimental and numerical measurements. The developed phase lag model is then coupled with the semi-analytical model of Lever and Weaver to predict the fluidelastic stability threshold. Improved predictions of the stability boundaries for the parallel triangular array were achieved. In addition, the present study has explained why fluidelastic instability does not occur below some threshold reduced velocity.