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.     

Propagation of a Tollmien-Schlichting wave over the junction between rigid and compliant surfaces

The propagation of an instability wave over the junction region between rigid and compliant panels is studied theoretically. The problem is investigated using three different methods with reference to flow in a plane channel containing sections with elastic walls. Within the framework of the first approach, using the solution of the problem of flow receptivity to local wall vibration, the problem considered is reduced to the solution of an integro-differential equation for the complex wall oscillation amplitude. It is shown that at the junction of rigid and elastic channel walls the instability-wave amplitude changes stepwise. For calculating the step value, another, analytical, method of investigating the perturbation propagation process, based on representing the solution as a superposition of modes of the locally homogeneous problem, is proposed. This method is also applied to calculating the flow stability characteristics in channels containing one or more elastic sections or consisting of periodically alternating rigid and compliant sections. The third method represents the unknown solution as the sum of a local forced solution and a superposition of orthogonal modes of flow in a channel with rigid walls. The latter method can be used for calculating the eigenvalues and eigenfunctions of the stability problem for flow in a channel with uniformly compliant walls.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 31–48. Original Russian Text Copyright © 2004 by Manuilovich.     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

Linear stability analysis of flow in a periodically grooved channel

We have conducted the linear stability analysis of flow in a channel with periodically grooved parts by using the spectral element method. The channel is composed of parallel plates with rectangular grooves on one side in a streamwise direction. The flow field is assumed to be two‐dimensional and fully developed. At a relatively small Reynolds number, the flow is in a steady‐state, whereas a self‐sustained oscillatory flow occurs at a critical Reynolds number as a result of Hopf bifurcation due to an oscillatory instability mode. In order to evaluate the critical Reynolds number, the linear stability theory is applied to the complex laminar flow in the periodically grooved channel by constituting the generalized eigenvalue problem of matrix form using a penalty‐function method. The critical Reynolds number can be determined by the sign of a linear growth rate of the eigenvalues. It is found that the bifurcation occurs due to the oscillatory instability mode which has a period two times as long as the channel period. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Linear instability characteristics of incompressible coaxial jets

The present study deals with the local linear instability of axisymmetric coaxial jets with a duct wall separating the two streams. The flow is assumed to be locally parallel, inviscid and incompressible. The objective of the work is to understand how the various parameters describing this flow geometry (i.e. boundary layers thicknesses at the exit, velocity ratio, wall thickness) may influence the instability of the flow and, in particular, the convective/absolute instability transition. A specific family of profiles is chosen for the modelling of the mean undisturbed flow and a spatial stability analysis is performed in order to identify the unstable modes and to assess how they are affected by the wake region behind the wall. An absolutely unstable mode is found, and its characteristics, depending on the velocity ratio and shear layers thicknesses, are determined. Results show that the absolute unstable mode is present only for a limited range of velocity ratios and that the corresponding frequency is almost constant if normalized with the mean velocity and wake thickness. This frequency value and the extension of the range of velocity ratios is similar to those found in the experiments on a similar geometry. Finally, a specific velocity ratio is found that maximizes the region at the jet exit for which an absolute instability behind the wall is present. This may increase the possibility for the onset of a global mode that may sustain the instability of the whole jet, enhancing considerably the mixing and entrainment characteristics between the two streams.     

Global modes in a confined impinging jet: application to heat transfer and control

We investigate the stability and control of a plane, laminar jet impinging on a flat plate in a channel, a geometry used to cool down a hot wall with a cold air jet in many industrial configurations. The global stability analysis indicates that, even for a strong confinement, the two-dimensional (2-D) steady flow is unstable to three-dimensional (3-D), steady perturbations. In the simplest limit case where dilatation effects are neglected, we show that the development of the instability induces a significant spanwise modulation of the heat flux at the impacted wall. To control the leading global mode, we propose adjoint-based 3-D harmonic and 2-D steady forcing in the bulk or at the wall. We show for instance that the unstable mode is controllable using a spanwise uniform blowing at the upper wall, in a specific domain corresponding to the footprint of the upper recirculating bubble. These techniques are applied to a novel open-loop control, in which we introduce into the flow a small airfoil, modelled by the lift force it exerts on the flow.     

Secondary instability modes generated by a Tollmien-Schlichting wave scattering from a bump

The plane finite-amplitude Tollmien-Schlichting wave interaction with a three-dimensional bump on a wall is considered for plane channel flow. The scattering of this wave leads to the production of unsteady three-dimensional disturbances which transform into growing secondary instability modes. The generation of such modes is studied assuming the three-dimensional disturbances to be small in comparison with the primary plane instability wave. The solution predicts that secondary disturbance amplification takes place only within a narrow wedge downstream of the bump. The qualitative comparison of results with experiments on turbulent wedge origination at an isolated roughness in a boundary layer is presented.     

Dynamics of rigid dumbbells in confined geometries Part II. Time-dependent shear flow

This paper describes analytical solutions to the response of a rigid dumbbell to a time-dependent simple shear flow in a channel of arbitrary size. The results were applied for specific flows, such as the sudden inception and cessation of simple shear and oscillatory shear flow. The introduction of confining walls leads to several qualitative differences compared with an unbounded flow such as initial negative jumps in the contribution of the dumbbell to the viscosity and the first normal stress coefficient. In addition, wall effects lead to a breakdown of well-known relationships between the complex viscosity and the complex normal stress coefficient during oscillatory flow.     

Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Based on a modified Darcy–Brinkman–Maxwell model, a linear stability analysis of a Maxwell fluid in a horizontal porous layer heated from below by a constant flux is carried out. The non-oscillatory instability and oscillatory instability with different hydrodynamic boundaries such as rigid and free surfaces at the bottom are studied. Compared with the rigid surface cases, onset of fluid motion due to non-oscillatory instability and oscillatory instability is found to occur both more easily for the system with a free bottom surface. The critical Rayleigh number for onset of fluid motion due to non-oscillatory instability is lower with a constant flux heating bottom than with an isothermal heating bottom, but the result due to oscillatory instability is in contrast. The effects of the Darcy number, the relaxation time, and the Prandtl number on the critical Rayleigh number are also discussed.     

Non-Local Interactions and Feedback Instability in a High Reynolds Number Flow

The question of non-locality is considered for a model supersonic flow at high Reynolds number in a channel formed between two parallel plates of different length, using the channel length as a control parameter. Examples are given of time-periodic stable and unstable flows forced by a disturbance positioned in the middle of the channel. It is shown that in certain parameter ranges the flow in a channel of ever increasing length is not approximated by the solutions obtained for infinitely long channels. This is interpreted in terms of a feedback interaction between the flow near the channel ends and the disturbance source. Feedback is shown to result from a slow upstream decay of disturbances coupled with a relatively fast downstream growth of instability waves. For a free (non-forced) flow, the feedback is found to lead to a form of global or resonant instability. Examples of growth rate calculations for the feedback modes are given.     

Interaction of gap flow with flapping dynamics of two side-by-side elastic foils

Two side-by-side elastic foils placed in an axial flow with the leading edges clamped lose their stability to exhibit in-phase or out-of-phase modes due to the proximity induced effects. Of particular, the passive out-of-phase flapping mode typically represents the clapping mechanism exhibited by biological organisms such as jellyfish and squid for swimming via jet propulsion. An impact of the viscous gap-flow dynamics on such passive flapping modes and vice versa is not well understood for the side-by-side elastic foil system. In the present work, we explore the mutual interaction of two side-by-side elastic foils performing flapping motion with the viscous gap-flow via a high-order finite element based fluid-elastic formulation with an exact tracking of fluid-foil interface. We show that the gap-flow exhibits pulsating flow with higher net drag for the passive out-of-phase coupled mode compared to the in-phase flapping where it exhibits uniform flow rate. Three distinct gap-flow velocity patterns are identified as functions of the coupled flapping modes: (i) unsteady symmetrical gap-flow with variable gap for the out-of-phase, (ii) unsteady alternating biased asymmetrical gap-flow with a uniform gap for the in-phase, and (iii) unsteady alternating biased asymmetrical gap-flow with variable gap for the mixed in-phase and out-of-phase. We examine the role of the gap-flow on the coupled fluid-elastic instability and the passive flapping modes. Two side-by-side elastic foils can experience significantly lower drag compared to their single foil counterpart and the two side-by-side rigid foils by undergoing static outward deformation. We utilize this phenomenon to understand the greater propensity of the flapping instability of the two side-by-side elastic foils in contrast to their single foil counterpart. We show that the coupled system does not exhibit the out-of-phase flapping if there is no gap-flow between the foils. We also find that two elastic foils when placed in proximity to each other always lose their stability to exhibit the out-of-phase coupling irrespective of whether the fully developed flapping exhibits in-phase or the out-of-phase flapping. The transition from the initial out-of-phase to the in-phase flapping is characterized by the loss of symmetry in the jet-like gap flow at the exit area of the side-by-side foils.     

Vibrations and stability of a periodically supported rectangular plate immersed in axial flow

Vibrations and stability of a thin rectangular plate, infinitely long and wide, periodically supported in both directions (so that it is composed by an infinite number of supported rectangular plates with slope continuity at the edges) and immersed in axial liquid flow on its upper side is studied theoretically. The flow is bounded by a rigid wall and the model is based on potential flow theory. The Galerkin method is applied to determine the expression of the flow perturbation potential. Then the Rayleigh–Ritz method is used to discretize the system. The stability of the coupled system is analyzed by solving the eigenvalue problem as a function of the flow velocity; divergence instability is detected. The convergence analysis is presented to determine the accuracy of the computed eigenfrequencies and stability limits. Finally, the effects of the plate aspect ratio and of the channel height ratio on the critical velocity giving divergence instability and vibration frequencies are investigated.     

Three-dimensionality of grooved channel flows at intermediate Reynolds numbers

 This paper describes the three-dimensional flow structure in grooved channels with different cavity lengths at intermediate Reynolds numbers. For steady flow, the three-dimensional effects are dominant near the side walls of the channel. However, after the onset of self-sustained oscillatory flow due to Tollmien–Schlichting waves as the primary instability, a secondary instability produces a three-dimensional flow with Taylor–Geortler-like vortical structure, at the bottom of the groove. This trend becomes more significant as the cavity length increases. Furthermore, the reason for three-dimensional flow is discussed using additional numerical analysis, and it is confirmed that the source of three-dimensional instability is the groove vortices due to the presence of side walls, rather than the channel traveling wave. Received: 7 September 1999/Accepted: 11 November 2000     

Stability Analysis of a Vertical Curved Channel Flow with a Radial Temperature Gradient

The linear stability analysis of a Newtonian incompressible fluid in a vertical curved channel formed by two coaxial cylindrical surfaces with a radial temperature gradient and an azimuthal pressure gradient shows that critical modes are oscillatory and non-axisymmetric. We have derived a generalized Rayleigh discriminant which includes both the curvature and buoyancy effects. Centrifugal buoyancy induces weak asymmetry of the dependence of the control parameter critical values on the sign of the temperature gradient. The critical parameters depend on the temperature gradient, the radius ratio and the nature of the fluid. For a wide curvature channel flow, there are two critical modes: oscillatory Dean modes for small temperature gradients and oscillatory centrifugal-thermal modes for relatively large temperature gradients. Received 14 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 Communicated by H.J.S. Fernando     

Absolute, convective, and global instability of a magnetic liquid jet

An infinite or semi-infinite jet of non-conductive magnetic liquid in a uniform longitudinal magnetic field can be absolutely or convectively unstable for different values of the flow parameters. Though the higher field inhibits the absolute instability, this inhibition is maximum at some field intensity. A critical value of the surface tension exists, above which the instability is absolute for any intensity of the field. If the jet has a large but finite length and proper boundary conditions are held at its beginning and end, it is always globally unstable. The unstable global mode is based on a pair of waves that propagate in opposite directions and reflect from one into the other at the flow boundaries.     

VIBRATION AND STABILITY OF VERTICAL UPWARD-FLUID-CONVEYING PIPE IMMERSED IN RIGID CYLINDRICAL CHANNEL

A theoretical model is developed for the vibration and stability of a vertical pipe subjected concurrently to two dependent axial flows. The external fluid, after exiting the outer annular region between the pipe and a rigid cylindrical channel, is conveyed upwards inside the pipe. This configuration thus resembles of a pipe that aspirating fluid. The equation of planar mo- tion is solved by means of the differential quadrature method （DQM）. Calculations are conducted for a slender drill-string-like and a bench-top-size system, for different confinement conditions of the outer annular channel. It is shown that the vibrations of these two systems are closely related to the degree of confinement of the outer annular channel. For a drill-string-like system with narrow annuli, buckling instability may occur in the second and third modes. For a bench-top-size system, however, both buckling and flutter may occur in the lowest three modes. The form of instability depends on the annuli size.     

Numerical calculation of two-dimensional flow in the channel with a partially compliant wall

The present paper is concerned with the flow in a two-dimensional channel whose wall is partially compliant. The flow field is calculated by the finite-difference method. Results are as follows: (1) When the upstream condition is given by steady flow (Reynolds number Re = 50), a compliant part of the wall oscillates with a frequency nearly equal to the characteristic frequency of the elastic wall. Absolute values of the pressure drop across the compliant part become small compared with those of the plane Poiseuille flow with wholly rigid walls. This ensures under physiological conditions that the blood can be transported more easily toward distal parts due to the compliance of vessel walls. (2) When the upstream condition is given by a pulsatile flow (Womersley number α = 8), interaction arises between characteristic frequency of the wall and basic frequency of the main stream near the compliant wall. As the basic frequency of pulsatile flow decreases, absolute values of mean pressure, which drop across the compliant wall, also become small compared with those of pulsatile flow between wholly rigid walls.     

TEMPORAL STABILITY OF FLOW THROUGH VISCOELASTIC TUBES

The stability of the Hagen–Poiseuille flow of a Newtonian fluid in an incompressible, viscoelastic tube contained within a rigid, hollow cylinder is determined using linear stability analysis. The stability of the system subjected to infinitesimal axisymmetric or non-axisymmetric disturbances is considered. The fluid and wall inertia terms are retained in their respective equations of motion. A novel numerical strategy is introduced to study the stability of the coupled fluid–structure system. The strategy alleviates the need for aninitial guess and thus ensures that all the unstable modes within a given closed region in the complex eigenvalue plane will be found. It is found that the system is unstable to both axisymmetric and non-axisymmetric disturbances. Moreover, depending on the values of the control parameters, the first unstable mode can be either an axisymmetric mode with the azimuthal wavenumber n=0 or a non-axisymmetric mode withn =1. For a given azimuthal wavenumber, it is found that there are no more than two unstable modes within the closed region considered here in the complex plane. For both the axisymmetric and non-axisymmetric instabilities, one mode is a solid-based, flow-induced surface instability, while the other one is a fluid-based instability that asymptotes to the least-damped rigid-wall mode as the thickness of the compliant wall tends to zero. All four modes are stabilized, to different degrees, by the solid viscosity.     

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.