各向异性柔性壁上二维T-S波演化的数值研究

受自然界鸟类羽毛的柔性特征启发, 利用数值模拟的手段进行了各向异性柔性壁面对亚音速边界层中T-S(Tollmien-Schlichting)波空间演化的影响研究. 首先, 刚性壁面上的数值结果与线性理论预测的结果吻合得很好, 验证了所采用的高阶精度格心型有限差分方法的可靠性. 在此基础上, 将部分刚性壁面替换为柔性壁面, 结果表明柔性壁面能够减小甚至消除T-S波的不稳定增长区间, 即抑制T-S波的发展, 因而具有推迟边界层转捩的潜力. 柔性壁面的变形不仅有对应T-S波波形的成分, 还会因柔性段前缘引起波长更长, 与T-S波频率相同的壁面波动. 随后开展的参数研究表明, 增大壁面阻尼削弱了前缘引起的壁面波动; 增大壁面的刚度、张力以及弹性系数都会使得壁面的刚性增强, 整体变形幅度下降; 柔性壁面的支撑杠杆臂倾角越大, 壁面刚性越强. 以上参数的增大均会使得柔性壁面抑制T-S波的效果降低. 此外, 当流动反方向流过时, 抑制T-S波的效果也会明显下降. 这些研究结果旨在揭示鸟类高效飞行的部分奥秘, 为被动减阻提供新的思路.      

Stability analysis of the onset of vortex shedding for wakes behind flat plates

Above a critical Reynolds number, wake flows behind flat plates become globally unstable, the leading modal instability in this case is known as Kelvin–Helmholtz mechanism. In this article, both local and BiGlobal linear instability analyses are performed numerically to study the onset of the shedding process. Flat plates with different base shapes are considered to assess geometry effects, and the relation between the critical shedding Reynolds number, \(Re_\mathrm{cr}\), and the boundary layer thickness is studied. Three types of base shapes are used: square, triangular and elliptic. It is found that the base shape has a great impact on the growth rate of least stable disturbance mode, thus would influence \(Re_\mathrm{cr}\) greatly, but it has little effect on the vortex shedding frequency. The shedding frequency is determined mainly by boundary layer thickness and has little dependence on the Reynolds number and base shape. We find that for a fixed Reynolds number, increasing boundary layer thickness acted in two ways to modify the global stability characteristics: It increases the length of the absolute unstable region and it makes the flow less locally absolutely unstable in the near-wake region, and these two effects work against each other to destabilize or stabilize the flow.     

On two-dimensional linear waves in Blasius boundary layer over viscoelastic layers

This work concerns the direct numerical simulation of small-amplitude two-dimensional ribbon-excited waves in Blasius boundary layer over viscoelastic compliant layers of finite length. A vorticity-streamfunction formulation is used, which assures divergence-free solutions for the evolving flow fields. Waves in the compliant panels are governed by the viscoelastic Navier's equations. The study shows that Tollmien–Schlichting (TS) waves and compliance-induced flow instability (CIFI) waves that are predicted by linear stability theory frequently coexist on viscoelastic layers of finite length. In general, the behaviour of the waves is consistent with the predictions of linear stability theory. The edges of the compliant panels, where abrupt changes in wall property occur, are an important source of waves when they are subjected to periodic excitation by the flow. The numerical results indicate that the non-parallel effect of boundary-layer growth is destabilizing on the TS instability. It is further demonstrated that viscoelastic layers with suitable properties are able to reduce the amplification of the TS waves, and that high levels of material damping are effective in controlling the propagating CIFI.     

ABSOLUTE AND CONVECTIVE BENDING INSTABILITIES IN FLUID-CONVEYING PIPES

The effect of internal plug flow on the lateral stability of fluid conveying pipes is investigated by determining the absolute or convective nature of the instability from the analytically derived linear dispersion relation. The fluid–structure interaction is modelled by following the work of Gregory & Paı̈doussis. The formulation of the fluid-conveying pipe problem is shown to be related to previous studies of a flat plate in the presence of uniform flow by Brazier-Smith & Scott and Crigthon & Oswell. The different domains of stability, convective instability, and absolute instability are explicitly derived in control parameter space. The effects of flow velocity, fluid–structure mass ratio, stiffness of the elastic foundation, bending rigidity and axial tension are considered. Absolute instability in flexural pipes prevails over a wide range of parameters. Convective instability is mostly found in tensioned pipes, which are modelled by a generalized linear Klein–Gordon equation. The impulse response is given in closed form or as an integral approximation and its behaviour confirms the results found directly from the dispersion equation.     

NUMERICAL SIMULATION OF STEADY FLOW IN A COMPLIANT TUBE OR CHANNEL WITH TAPERED WALL THICKNESS

Flow through compliant tubes with linear taper in wall thickness is numerically simulated by finite element analysis. Two models are examined: a compliant channel and an axisymmetric tube. For verification of the numerical method, flow through a compliant stenotic vessel is simulated and compared to existing experimental data. Steady two-dimensional flow in a collapsible channel with initial tension is also simulated and the results are compared with numerical solutions from the literature. Computational results for an axisymmetric tube show that as cross-sectional area falls with a reduction in downstream pressure, flow rate increases and reaches a maximum when the speed index (mean velocity divided by wave speed) is near unity at the point of minimum cross-sectional area, indicative of wave-speed flow limitation or “choking” (flow speed equals wave speed) in previous one-dimensional studies. For further reductions in downstream pressure, the flow rate decreases. Cross-sectional narrowing is significant but localized. For the particular wall and fluid properties used in these simulations, the area throat is located near the downstream end when the ratio of downstream-to-upstream wall thickness is 2; as wall taper is increased to 3, the constriction moves to the upstream end of the tube. In the planar two-dimensional channel, area reduction and flow limitation are also observed when outlet pressure is decreased. In contrast to the axisymmetric case, however, the elastic wall in the two-dimensional channel forms a smooth concave surface with the area throat located near the mid-point of the elastic wall. Though flow rate reaches a maximum and then falls, the flow does not appear to be choked.     

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.     

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     

Nonlinear dynamics, stability and control of the scan process in noncontacting atomic force microscopy

The nonlinear equations of motion for the scan process in noncontacting atomic force microscopy are consistently derived using the extended Hamilton’s principle. A modal dynamical system obtained from the continuum model reveals that scan control appears in the form of parametric excitation. The system is analyzed asymptotically and numerically to yield escape bounds limiting the noncontacting mode of operation. Approximate stability bounds are deduced from both a global Melnikov integral and a local Moon–Chirikov overlap criterion. The Melnikov–Holmes stability curve and the overlap criterion are found to be similar for small damping. However, for very small damping, typical of ultra-high vacuum conditions, where the Melnikov bound becomes trivial, the Moon–Chirikov criterion yields an improved stability threshold.     

Experimental and theoretical observations of elastic instabilities in eccentric cylinder flows: local versus global instability

A theoretical and experimental investigation of the stability of the viscoelastic flow of a Boger fluid between eccentric cylinders is presented. In our theoretical study, a local linear stability analysis for the flow of an Oldroyd-B fluid suggests that the flow is elastically unstable for all eccentricities. A global solution to the stability problem is obtained by a perturbation eigenvalue analysis, incorporating the azimuthal variation of the base state flow at the same order as the streamwise variation of the stability function. A comparison between the local and global stability predictions is made. Flow visualization experiments with a solution of high molecular weight polyisobutylene dissolved in a viscous solvent clearly show the transition from a purely azimuthal flow to a secondary toroidal flow. Comparison of these experimental results with the local linear stability theory shows good agreement between the measured and predicted critical conditions for the onset of the non-inertial cellular instability at small δ, where δ is the eccentricity made dimensionless with the average gap thickness. At higher eccentricities, experiment and local linear stability theory cease to agree. Evidence will be given that this disagreement is due to a global affect, i.e. the convection of stress not included the local theory. Specifically, it is suggested that convection of polymeric stresses in the base flow as well as in the disturbance flow can stabilize the instabilities found in this geometry. Finally, the discovery of a new localized purely elastic instability associated with the recirculation flow in the co-rotating eccentric cylinder geometry is presented.     

Long waves between a subsonic gas flow-film interface flowing down an inclined plate

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of a parallel subsonic gas flow. The waves are described by evolution equation previously derived as a generalization of the model for the Newtonian liquid. We confirm linear stability results of the basic flow using the Orr–Sommerfeld analysis to that obtained by long wave approximation analysis. The non-linear stability criteria of the model are discussed analytically and stability branches are obtained. Finally, the solitary wave solutions at the liquid–gas interface are discussed, using specially envelope transform and direct ansatz approach to Ginzburg–Landau equation. The influence of different parameters governing the flow on the stability behavior of the system is discussed in detail.     

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.     

Global stability behaviour for the BEK family of rotating boundary layers

Numerical simulations were conducted to investigate the linear global stability behaviour of the Bödewadt, Ekman, von Kármán (BEK) family of flows, for cases where a disc rotates beneath an incompressible fluid that is also rotating. This extends the work reported in recent studies that only considered the rotating-disc boundary layer with a von Kármán configuration, where the fluid that lies above the boundary layer remains stationary. When a homogeneous flow approximation is made, neglecting the radial variation of the basic state, it can be shown that linearised disturbances are susceptible to absolute instability. We shall demonstrate that, despite this prediction of absolute instability, the disturbance development exhibits globally stable behaviour in the BEK boundary layers with a genuine radial inhomogeneity. For configurations where the disc rotation rate is greater than that of the overlying fluid, disturbances propagate radially outwards and there is only a convective form of instability. This replicates the behaviour that had previously been documented when the fluid did not rotate beyond the boundary layer. However, if the fluid rotation rate is taken to exceed that of the disc, then the propagation direction reverses and disturbances grow while convecting radially inwards. Eventually, as they approach regions of smaller radii, where stability is predicted according to the homogeneous flow approximation, the growth rates reduce until decay takes over. Given sufficient time, such disturbances can begin to diminish at every radial location, even those which are positioned outwards from the radius associated with the onset of absolute instability. This leads to the confinement of the disturbance development within a finitely bounded region of the spatial–temporal plane.     

Free-edge stresses and progressive cracking in surface coatings of circular torsion bars

This paper considers the explicit solutions of free-edge stresses near circumferential cracks in surface coatings of circular torsion bars and their application in determining the progressive cracking density in the coating layers. The problem was formulated within the framework of linear elastic fracture mechanics (LEFM). The free-edge stresses near crack tip and the shear stresses in the cross-section of the torsion bar were approached in explicit forms based on the variational principle of complementary strain energy. Criterion for progressive cracking in the coating layer was established in sense of strain energy conservation, and the crack density is thereby estimated. Effects of external torque, aspect ratio, and elastic properties on the density of progressive cracking were examined numerically. The present study shows that, in the sense of inducing a given crack density, compliant coating layer with lower modulus has much higher critical torque than that of a stiffer one with the same geometries and substrate material, i.e., compliant coating layer has greater cracking tolerance. Meanwhile, the study also indicates that thicker surface coating layer is more pliant to cracking than the thinner ones. The present model can be used for analyzing the damage mechanism and cracking tolerance of surface coatings of torsion shafts and for data reduction of torsional fracture test of brittle surface coatings, etc.     

Global stability of separated flows: subsonic flow past corners

The triple-deck equations for the steady subsonic flow past a convex corner are solved numerically using a novel technique based on Chebychev collocation in the direction normal to the body combined with finite differences in the direction along the flow. The resulting set of nonlinear algebraic equations are solved with Newton linearization and using the GMRES method for the solution of the linear system of equations. The stability of the computed steady flows is then examined using global stability analysis. It is found that for small corner angles, the Tollmien?CSchlichting modes are globally unstable and these persist to larger corner angles. Multiple steady state solutions also exist beyond a critical corner angle but these are globally unstable because of the presence of the Tollmien?CSchlichting modes.     

Sound propagation through a rarefied gas. Influence of the gas–surface interaction

Acoustic waves propagating through a rarefied gas between two plates induced by both oscillation and unsteady heating of one of them are considered on the basis of a model of the linearized Boltzmann equation. The gas flow is considered as fully established so that the dependence of all quantities on time is harmonical. The problem is solved for several values of two main parameters determining its solution, namely, the gas rarefaction defined as the ratio of the distance between the plates to the equivalent free path of gaseous molecules, and the oscillation parameter given as the ratio of the intermolecular collision frequency to the wave frequency. The reciprocal relation for such flows is obtained and verified numerically. An influence of the gas–surface accommodation coefficients on the wave characteristics is analyzed by employing the Cercignani–Lampis scattering kernel to the boundary conditions.     

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.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.     

On the General Ericksen–Leslie System: Parodi’s Relation, Well-Posedness and Stability

In this paper we investigate the role of Parodi’s relation in the well-posedness and stability of the general Ericksen–Leslie system modeling nematic liquid crystal flows. First, we give a formal physical derivation of the Ericksen–Leslie system through an appropriate energy variational approach under Parodi’s relation, in which we can distinguish the conservative/dissipative parts of the induced elastic stress. Next, we prove global well-posedness and long-time behavior of the Ericksen–Leslie system under the assumption that the viscosity μ ₄ is sufficiently large. Finally, under Parodi’s relation, we show the global well-posedness and Lyapunov stability for the Ericksen–Leslie system near local energy minimizers. The connection between Parodi’s relation and linear stability of the Ericksen–Leslie system is also discussed.     

NUMERICAL SIMULATION OF LINEAR AND NONLINEAR DISTURBANCE EVOLUTION IN A BOUNDARY LAYER WITH COMPLIANT WALLS

A computational method capable of simulating the spatial evolution of disturbances in a boundary-layer flow with compliant coatings has been developed. The flow geometry being an unknown of the problem, this difficulty is overcome by use of a mapping, the domain being fixed in the computational coordinates. The model takes into account the nonlinear fluid-structure interaction all over the flow field, as well as nonparallel effects due to the wall displacement and to the boundary-layer growth. First, the numerical solution procedure is tested by focusing on the linear and nonlinear spatial disturbance evolution for a spring-backed elastic plate which is only unstable with respect to Tollmien–Schlichting-type travelling waves. The numerical procedure is then used to study the influence of the initial disturbance amplitude on the disturbance development for a tensioned membrane. Finally, to simulate a true physical experiment, a spring-backed elastic plate of finite length is considered. It is shown that the numerical model is capable of detecting the interaction between Tollmien–Schlichting instabilities and flow-induced surface instabilities at the interface.     

Effect of Volume Energy Supply on the Stability of a Subsonic Vortex Flow

Calculations of the stability of an axisymmetric vortex flow of viscous heat-conducting gas with volume energy supply are presented. The unperturbed axisymmetric vortex flow was found numerically using a quasi-cylindrical approximation of the Navier-Stokes equations under the assumption of constant peripheral-velocity circulation in the ambient co-current flow. The volume energy supply in the viscous vortex core was modeled by an additional source term in the energy equation. The stability characteristics of the viscous vortex flow in a longitudinal vortex with respect to both axisymmetric and non-axisymmetric three-dimensional waves traveling along the vortex axis and corresponding to both positive and negative values of the azimuthal wave number were found using the time-dependent formulation of the linear stability theory for compressible three-dimensional plane-parallel flows.