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.     

Aeroelastic stability of a cantilevered plate in yawed subsonic flow

The aeroelastic stability of cantilevered plates with their clamped edge oriented both parallel and normal to subsonic flow is a classical fluid–structure interaction problem. When the clamped edge is parallel to the flow the system loses stability in a coupled bending and torsion motion known as wing flutter. When the clamped edge is normal to the flow the instability is exclusively bending and is referred to as flapping flag flutter. This paper explores the stability of plates during the transition between these classic aeroelastic configurations. The aeroelastic model couples a classical beam structural model to a three-dimensional vortex lattice aerodynamic model. The aeroelastic stability is evaluated in the frequency domain and the flutter boundary is presented as the plate is rotated from the flapping flag to the wing configuration. The transition between the flag-like and wing-like instability is often abrupt and the yaw angle of the flow for the transition is dependent on the relative spacing of the first torsion and second bending natural frequencies. This paper also includes ground vibration and aeroelastic experiments carried out in the Duke University Wind Tunnel that confirm the theoretical predictions.     

Flutter of spring-mounted flexible plates in uniform flow

A fluid–structure interaction (FSI) system is studied wherein a cantilevered flexible plate aligned with a uniform flow has its upstream end attached to a spring mounting. This allows the entire system to oscillate in a direction perpendicular to that of the flow as a result of the mounting׳s dynamic interaction with the flow-induced oscillations, or flutter, of the flexible plate. We also study a hinged-free rotational-spring attachment as a comparison for the heaving system. This variation on classical plate flutter is motivated by its potential as an energy-harvesting system in which the reciprocating motion of the support system would be tapped for energy production. We formulate and deploy a hybrid of theoretical and computational modelling for the two systems and comprehensively map out their linear-stability characteristics at low mass ratio. Relative to a fixed cantilever, the introduction of the dynamic support in both systems yields lower flutter-onset flow speeds; this is desirable for energy-harvesting applications. We further study the effect of adding an inlet surface upstream of the mount as a means of changing the destabilising mechanism from single-mode flutter to modal-coalescence flutter which is a more powerful instability more suited to energy harvesting. This strategy is seen to be effective in the heaving system. However, divergence occurs in the rotational system for low spring natural frequencies and this would lead to its failure for energy production. Finally, we determine the power-output characteristics for both systems by introducing dashpot damping at the mount. The introduction of damping increases the critical speeds and its variation permits optimal values to be found that maximise the power output for each system. The addition of an inlet surface is then shown to increase significantly the power output of the heaving system whereas this design strategy is not equally beneficial for the rotational system.     

Modal analysis of cantilever plate flutter

This study illustrates the mechanism of modal coupling in cantilever plate flutter using the full Theodorsen airfoil theory within the linear framework. An accurate, pseudo-spectral method is employed to calculate the fluid loading and the eigenvalue problem is solved numerically following the Galerkin procedure. For plates with a structure-to-fluid mass ratio around unity, the first two in vacuo modes are dominant and the Kutta condition at the trailing edge plays a central role in the flutter mechanism. The fluid loading induced by the first mode excites significant second and higher order modes. The fluid loading on the second mode is coupled strongly with the structural vibration velocity of the first mode, which is identified as the main mechanism of energy transfer from flow to plate. It is demonstrated that the response of the second mode is suppressed and the plate is stabilized when a concentrated mass is added near the middle of the plate length. Theoretical prediction is supported by experimental data although the latter is affected by many practical factors that are difficult to model precisely.     

Flutter of an elastic plate in a channel flow: Confinement and finite-size effects

When a cantilevered plate lies in an axial flow, it is known to exhibit self-sustained oscillations once a critical flow velocity is reached. This flutter instability has been investigated theoretically, numerically and experimentally by different authors, showing that the critical velocity is always underestimated by two-dimensional models. However, it is generally admitted that, if the plate is confined in the spanwise direction by walls, three-dimensionality of the flow is reduced and the two-dimensional models can apply. The aim of this article is to quantify this phenomenon by analysing the effect of the clearance between the plate and the side walls on the flutter instability. To do so, the pressure distribution around an infinite-length plate is first solved in the Fourier space, which allows to develop an analytical model for the pressure jump. This model is then used in real space to compute instability thresholds as a function of the channel clearance, the plate aspect ratio and mass ratio. Our main result shows that, as the value of the clearance is reduced, the convergence towards the two-dimensional limit is so slow that this limit is unattainable experimentally.     

Numerical investigation of the effect of boundary conditions on hydroelastic stability of two parallel plates interacting with a layer of ideal flowing fluid

The paper studies the hydroelastic stability of two parallel identical rectangular plates interacting with a flowing fluid confined between them. General equations describing the behavior of ideal compressible liquid in the case of small perturbations are written in terms of the perturbation velocity potential and transformed using the Bubnov–Galerkin method. The small deformations of elastic plates are defined using the first-order shear deformation plate theory. A mathematical formulation of the dynamic problem for elastic structures is developed using the variational principle of virtual displacements, which takes into account the work done by the inertial forces and hydrodynamic pressure. The numerical solution of the problem is carried out in three-dimensional formulation by means of the finite element method. A stability criterion is based on the analysis of complex eigenvalues of the coupled system of equations obtained for different values of flow velocity. The existence of different types of instability has been shown depending on the combinations of the kinematic boundary conditions defined at the edges of both plates. We considered both the symmetric and asymmetric types of clamping. It has been found that the dependence of the lowest eigenfrequency of two parallel plates on the height of quiescent fluid is nonmonotonic with a pronounced peak. At the same time, critical velocities of instability change insignificantly if the distance between plates is greater than half of the maximum linear dimensions of the structure. It should be noted that the critical velocities of divergence increase monotonically with growth of the height of the fluid layer, but critical velocities for the onset of flutter instability have sharp jumps. The cause of these jumps is a change in the mode shapes at which the system loses stability.     

The effect of inertial inhomogeneity on the flutter of a cantilevered flexible plate

We study the effect of adding discrete structural mass on the linear stability of an otherwise homogeneous cantilevered-free flexible plate immersed in uniform axial flow. The methods of Howell et al. that mixed numerical simulation with eigenvalue analysis are simply extended for the present study. An ideal two-dimensional flow is assumed wherein the rotationality of the boundary-layers is modelled by vortex elements on the solid-fluid interface and the imposition of the Kutta condition at the plate's trailing edge. The Euler-Bernoulli beam model is used for the structural dynamics. It is shown that addition of mass to the plate can be either stabilising or destabilising, depending upon the location of the added mass, and how its inclusion modifies the energy exchanges of the corresponding homogeneous structure. Our results therefore suggest a straightforward means by which the critical flow speed at which low-amplitude flutter sets in can be passively controlled in engineering applications.     

On bimodal flutter behavior of a flexible airfoil

The dynamic aeroelastic behavior of an elastically supported airfoil is studied in order to investigate the possibilities of increasing critical flutter speed by exploiting its chord-wise flexibility. The flexible airfoil concept is implemented using a rigid airfoil-shaped leading edge, and a flexible thin laminated composite plate conformally attached to its trailing edge. The flutter behavior is studied in terms of the number of laminate plies used in the composite plate for a given aeroelastic system configuration. The flutter behavior is predicted by using an eigenfunction expansion approach which is also used to design a laminated plate in order to attain superior flutter characteristics. Such an airfoil is characterized by two types of flutter responses, the classical airfoil flutter and the plate flutter. Analysis shows that a significant increase in the critical flutter speed can be achieved with high plunge and low pitch stiffness in the region where the aeroelastic system exhibits a bimodal flutter behavior, e.g., where the airfoil flutter and the plate flutter occur simultaneously. The predicted flutter behavior of a flexible airfoil is experimentally verified by conducting a series of systematic aeroelastic system configurations wind tunnel flutter campaigns. The experimental investigations provide, for each type of flutter, a measured flutter response, including the one with indicated bimodal behavior.     

Local and global instabilities of flow in a flexible-walled channel

We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling-wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability.     

Flutter of a rectangular plate

We address theoretically the linear stability of a variable aspect ratio, rectangular plate in a uniform and incompressible axial flow. The flutter modes are assumed to be two-dimensional but the potential flow is calculated in three dimensions. For different values of aspect ratio, two boundary conditions are studied: a clamped-free plate and a pinned-free plate. We assume that the fluid viscosity and the plate viscoelastic damping are negligible. In this limit, the flutter instability arises from a competition between the destabilising fluid pressure and the stabilising flexural rigidity of the plate. Using a Galerkin method and Fourier transforms, we are able to predict the flutter modes, their frequencies and growth rates. The critical flow velocity is calculated as a function of the mass ratio and the aspect ratio of the plate. A new result is demonstrated: a plate of finite span is more stable than a plate of infinite span.     

Flow field around the flapping flag

The flapping flag is a canonical fluid–structure interaction problem that describes a cantilever plate with flow along its elastic axis. When the flapping flag loses stability it enters a large amplitude Limit Cycle Oscillation (LCO). While theoretical models can accurately predict the flutter velocity and frequency, there are still discrepancies between the experimental observations and the theoretical predictions of the post-critical LCO response. This note provides recent flow field visualizations in a single longitudinal plane for a cantilevered aluminum plate in axial flow during its LCO. Particle Image Velocimetry (PIV) techniques are used to show that the flow over the midspan of the plate is attached even during the violent LCO motion. This observation suggests that potential flow aerodynamic models may be able to capture the essential features in the flow field.     

A flexible rod in annular leakage flow: Influence of turbulence and equilibrium offset,and analysis of instability mechanisms

The linear stability of a flexible, cylindrical rod subjected to annular leakage flow is studied. The mathematical models developed by Li, Kaneko, and Hayama in 2002 and Fujita and Shintani in 2001 are bridged and extended, to account for a flexible rod with equilibrium offset (eccentricity) in laminar or turbulent leakage flow. Stability characteristics are analyzed numerically for a variety of configurations. It is found that simply supported rods may become unstable at a certain critical flow speed by either divergence or flutter, depending on dimensions and fluid/solid properties. It is furthermore found that the critical flow speed is quite insensitive to use of a laminar friction model at high Reynolds numbers in cases of divergence, but sensitive to it in cases of flutter. These findings are verified analytically though analysis of an energy equation. This equation shows that (i) divergence instability is independent of fluid friction; (ii) flutter instability is caused solely by fluid friction. It also suggests a possible explanation to the question of why a ‘wrong’ fluid friction assumption gives a too large critical flow speed in cases of flutter instability at a high Reynolds number.     

Flutter and resonances of a flag near a free surface

We investigate the effects of a nearby free surface on the stability of a flexible plate in axial flow. Confinement by rigid boundaries is known to affect flag flutter thresholds and fluttering dynamics significantly, and this work considers the effects of a more general confinement involving a deformable free surface. To this end, a local linear stability is proposed for a flag in axial uniform flow and parallel to a free surface, using one-dimensional beam and potential flow models to revisit this classical fluid–structure interaction problem. The physical behaviour of the confining free surface is characterized by the Froude number, corresponding to the ratio of the incoming flow velocity to that of the gravity waves. After presenting the simplified limit of infinite span (i.e. two-dimensional problem), the results are generalized to include finite-span and lateral confinement effects. In both cases, three unstable regimes are identified for varying Froude number. Rigidly-confined flutter is observed for low Froude number, i.e. when the free surface behaves as a rigid wall, and is equivalent to the classical problem of the confined flag. When the flow and wave velocities are comparable, a new instability is observed before the onset of flutter (i.e. at lower reduced flow speed) and results from the resonance of a structural bending wave and one of the fundamental modes of surface gravity waves. Finally, for large Froude number (low effect of gravity), flutter is observed with significant but passive deformation of the free surface in response of the flag’s displacement.     

Fluid structure interaction of cantilever micro and nanotubes conveying magnetic fluid with small size effects under a transverse magnetic field

Micro and nanotubes have found major application in fluidic systems as channels for conveying fluid. In some micro and nanofluidic applications such as drug delivery, a transverse magnetic field can be used to guide the fluid flow by generating an axial force in the flow direction. An important issue in the design of micro and nanofluidic systems is the structural vibration caused by the fluid flow. In the current study, we investigate the effect of transverse magnetic field on the vibration of cantilever micro and nanotubes conveying fluid by considering the small size effects. We couple the nonlocal Euler–Bernoulli beam model with Navier–Stokes theory to determine a fluid structure interaction (FSI) model for the vibration analysis of the system. We modify the FSI governing equation by driving a velocity correction factor to consider the effect of transverse magnetic field on the fluid flow’s pattern through the tube. Then, we use the Galerkin’s method to obtain the frequency diagrams for the instability analysis of the system. We show that the transverse magnetic field can have a substantial effect on the dynamics of tube conveying fluid by increasing the system’s natural frequencies and critical flow velocity which contributes to the flutter instability. We also discover that although the transverse magnetic field plays a crucial role on dynamics of microstructures, its effect on the dynamics of nanotubes is not significant and can be ignored.     

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.     

The dynamics of two-dimensional cantilevered plates with an additional spring support in axial flow

In this paper, the dynamics of two-dimensional cantilevered flexible plates in axial flow is investigated using a fluid–structure interaction model. An additional spring support of either linear or cubic type is installed at various locations on the plate; its presence qualitatively affects the dynamics of the fluid–structure system. Without the spring, the cantilevered plate loses stability by flutter when the flow velocity exceeds a critical value; as the flow velocity increases further, the system dynamics is qualitatively the same: the plate undergoes symmetric limit cycle oscillations with increasing amplitude. With a linear spring, a state of static buckling is added to the dynamics. Rich nonlinear dynamics can be observed when a cubic spring is considered; the plate may be stable and buckled, and it may undergo either symmetric or asymmetric limit cycle oscillations. Moreover, when the flow velocity is sufficiently high, the plate may exhibit chaotic motions via a period-doubling route.     

DEVELOPMENT OF A THREE-DIMENSIONAL VISCOUS AEROELASTIC SOLVER FOR NONLINEAR PANEL FLUTTER

A new three-dimensional (3-D) viscous aeroelastic solver for nonlinear panel flutter is developed in this paper. A well-validated full Navier–Stokes code is coupled with a finite-difference procedure for the von Karman plate equations. A subiteration strategy is employed to eliminate lagging errors between the fluid and structural solvers. This approach eliminates the need for the development of a specialized, tightly coupled algorithm for the fluid/structure interaction problem. The new computational scheme is applied to the solution of inviscid two-dimensional panel flutter problems for subsonic and supersonic Mach numbers. Supersonic results are shown to be consistent with the work of previous researchers. Multiple solutions at subsonic Mach numbers are discussed. Viscous effects are shown to raise the flutter dynamic pressure for the supersonic case. For the subsonic viscous case, a different type of flutter behavior occurs for the downward deflected solution with oscillations occurring about a mean deflected position of the panel. This flutter phenomenon results from a true fluid/structure interaction between the flexible panel and the viscous flow above the surface. Initial computations have also been performed for inviscid, 3-D panel flutter for both supersonic and subsonic Mach numbers.     

Analysis of the aerodynamic interaction between two plunging plates in tandem at low Reynolds number for maximum propulsive efficiency

The thrust generated by two heaving plates in tandem is analyzed computationally by solving the Navier–Stokes equations for an incompressible and two-dimensional flow at low Reynolds numbers. We consider with detail two particular sets of configurations of interest in forward flight in a wide range of heaving amplitudes and frequencies: a plunging leading plate with the trailing plate at rest, and the two plates heaving with the same frequency and amplitude, but varying the phase difference. In almost all cases the thrust efficiency of the leading plate is augmented in relation to a single plate heaving with the same frequency and amplitude. In the first configuration with a trailing plate at rest, we characterize the range of nondimensional heaving frequencies and amplitudes of the leading plate for which the stationary trailing plate contributes positively to the global thrust. The maximum global thrust efficiency of this configuration, reached for an advance ratio slightly less than unity and a reduced frequency close to 5, is about the same as the maximum efficiency for an isolated plate, reached for slightly smaller frequencies. But for low frequencies the tandem configuration with the trailing plate at rest is more thrust efficient than the isolated plate. We also characterize the nondimensional frequency and amplitude regions for which the flow becomes chaotic. In the second configuration, the maximum of the total thrust efficiency is reached for a phase lag of 180° (counterstroking), particularly for an advance ratio unity and a reduced frequency 4.4. It is almost the same as the maximum thrust efficiency in the other configuration with the trailing plate at rest and that of a single plate. We discuss the flow structures and the aerodynamic interaction between plates responsible for the optimal thrust configuration in both cases.     

磁场对不同温度场中输流悬臂碳纳米管动态特性的影响

本文在采用经典欧拉-伯努利梁模型的基础上,引入考虑小尺度效应的非局部弹性理论,着重研究不同温度场中输流悬臂单层碳纳米管系统（SWCNT）在外加纵向磁场作用下的颤振失稳问题。基于哈密顿原理获得了该流固耦合系统的振动控制方程及相应的边界条件,应用微分变换法（DTM法）求解此高阶偏微分方程,通过数值计算研究了不同温度场中施加纵向磁场对系统动力学特性的影响。结果表明：施加纵向磁场在不同温度场中都将增强输流悬臂碳纳米管的动态稳定性。然而,这种增强程度却与温度场的变化量有关,在不同温度变化量下,磁场对系统稳定性的增强程度有一个峰值,这意味着,实际应用中,为了提高这类流固耦合系统的动态稳定性,一味提高纵向磁场强度并不可取。     

Aeroelastic behavior of a flag in ground effect

The aeroelastic behavior of a flexible plate subjected to a uniform axial flow is investigated in the presence of a rigid plane set parallel to the plate. It is shown that the ground effect reduces the flutter inflow velocity and strengthens the possibility of using the flag for extracting energy from winds and currents. The numerical analysis is carried out assuming that both the unsteady potential incompressible flow and the plate can be described with 2D models, i.e., a lumped vortex panel method and a nonlinear Euler–Bernoulli beam model, respectively, without losing the essential features of the fluid–structure interaction. Asymmetry of post-critical behavior (limit-cycle oscillations) and sensitivity of the results to the main flag parameters (distance from the ground, mass ratio and damping) are also considered, including also the energy distribution over the identified proper orthogonal modes. The investigated reduction of the flutter velocity in ground effect has been also confirmed with experimental tests relative to a polypropylene flag with and without the rigid panel mimicking the presence of the ground.