考虑分布轴向力的细长杆横向振动与失稳分析

为量化梁、杆、柱的自重(下称分布轴向力)对静力失稳和动力横向振动的影响,在《材料力学》和《机械系统动力学》教材的基础上,建立了分布轴向力下的杆柱失稳和横向振动的力学、数学模型.采用有限差分法、伽辽金法和数值积分法获取计算结果.结果表明:考虑分布轴向力的杆柱横向振动固有频率随杆长增加而减小,杆柱失稳时一阶固有频率为0;分...     

The energy transfer mechanism of a perturbed solid-body rotation flow in a rotating pipe

Three-dimensional direct numerical simulations of a solid-body rotation superposed on a uniform axial flow entering a rotating constant-area pipe of finite length are presented. Steady in time profiles of the radial, axial, and circumferential velocities are imposed at the pipe inlet. Convective boundary conditions are imposed at the pipe outlet.The Wang and Rusak(Phys. Fluids 8:1007–1016, 1996.doi:10.1063/1.86882) axisymmetric instability mechanism is retrieved at certain operational conditions in terms of incoming flow swirl levels and the Reynolds number. However, at other operational conditions there exists a dominant,three-dimensional spiral type of instability mode that is consistent with the linear stability theory of Wang et al.(J. Fluid Mech. 797: 284–321, 2016). The growth of this mode leads to a spiral type of flow roll-up that subsequently nonlinearly saturates on a large amplitude rotating spiral wave. The energy transfer mechanism between the bulk of the flow and the perturbations is studied by the Reynolds-Orr equation. The production or loss of the perturbation kinetic energy is combined of three components: the viscous loss, the convective loss at the pipe outlet, and the gain of energy at the outlet through the work done by the pressure perturbation. Theenergy transfer in the nonlinear stage is shown to be a natural extension of the linear stage with a nonlinear saturated process.     

Application of the Dorodnitsyn finite element method to swirling boundary layer flow

The Dorodnitsyn finite element method for turbulent boundary layer flow with surface mass transfer is extended to include axisymmetric swirling internal boundary layer flow. Turbulence effects are represented by the two-layer eddy viscosity model of Cebeci and Smith¹ with extensions to allow for the effect of swirl. The method is applied to duct entry flow and a 10 degree included-angle conical diffuser, and produces results in close agreement with experimental measurements with only 11 grid points across the boundary layer. The introduction of swirl (w_e/u_e = 0.4) is found to have little effect on the axial skin friction in either a slightly favourable or adverse pressure gradient, but does cause an increase in the displacement area for an adverse pressure gradient. Surface mass transfer (blowing or suction) causes a substantial reduction (blowing) in axial skin friction and an increase in the displacement area. Both suction and the adverse pressure gradient have little influence on the circumferential velocity and shear stress components. Consequently in an adverse pressure gradient the flow direction adjacent to the wall is expected to approach the circumferential direction at some downstream location.     

The effects of inlet conditions and expansion ratio on the onset of flow reversal in swirling flow in a sudden expansion

Experiments are reported in which the minimum swirl intensity required to produce a central recirculation zone in a swirling sudden expansion flow is determined as a function of expansion ratio and inlet conditions. Using a swirl generator which allows for independent variation of velocity profile shape and swirl number, it is shown that an inlet tangential velocity distribution with a large solid body vortex core or an axial velocity profile with a maximum on the axis will lead to a higher critical swirl.     

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.     

Weakly non-linear waves in a fluid-filled elastic tube with variable prestretch

In the present work, by utilizing the non-linear equations of motion of an incompressible, isotropic thin elastic tube subjected to a variable prestretch both in the axial and the radial directions and the approximate equations of motion of an incompressible inviscid fluid, which is assumed to be a model for blood, we studied the propagation of weakly non-linear waves in such a medium, in the long wave approximation. Employing the reductive perturbation method we obtained the variable coefficient KdV equation as the evolution equation. By seeking a travelling wave solution to this evolution equation, we observed that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations.     

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.     

Time and spatially resolved measurements of interfacial waves in vertical annular flow

Film thickness measurements have been performed in a vertical air/water annular flow in a pipe of 0.05 m diameter. A sensor has been built which allows to measure the film thickness evolution in time at 320 positions, such that the interface of the vertical annular flow can be reconstructed. The large-scale structures moving on the interface are described statistically, with a special attention to the disturbance waves. Probability density functions and mean statistics are given for the height, length, velocity, frequency and spatial distribution of the disturbance waves. In particular, it is shown that the disturbance waves are three-dimensional structures with large height fluctuations in the circumferential and axial direction, giving a meandering path between the maximum height around the circumference. It is also shown that the disturbance waves can flow with a slight inclination with respect to the axial direction. Finally, the disturbance waves are shown to be located randomly in space, within a Gamma distribution whose order only depends on the liquid superficial velocity. Due to the nature of the Gamma distribution, it could indicate that the spatial distribution of the disturbance waves results from a cascade of coalescence processes between the original disturbance waves on the film.     

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.     

Nonlinear instability of an annular liquid sheet exposed to gas flow

Nonlinear instability and breakup of an annular liquid sheet has been modeled in this paper. The liquid sheet is considered to move axially and is exposed to co-flowing inner and outer gas streams. Also, the effect of outer gas swirl on sheet breakup has been studied. In the developed model a perturbation expansion method has been used with the initial magnitude of the disturbance as the perturbation parameter. This is a comprehensive model in that other geometries of planar sheet and a coaxial jet can be obtained as limiting cases of very large inner radius and inner radius equal to zero, respectively. In this temporal analysis, the effect of liquid Weber number, initial disturbance amplitude, inner gas-to-liquid velocity ratio, outer gas-to-liquid velocity ratio and outer gas swirl strength on the breakup time is investigated. The model is validated by comparison with earlier analytical studies for the limiting case of a planar sheet as well as with experimental data of sheet breakup length available in literature. It is shown that the linear theory cannot predict breakup of an annular sheet and the developed nonlinear model is necessary to accurately determine the breakup length. In the limiting case of a coaxial jet, results show that gas swirl destabilizes the jet, makes helical modes dominant compared to the axisymmetric mode and decreases jet breakup length. These results contradict earlier linear analyses and agree with experimental observations. For an annular sheet, it is found that gas flow hastens the sheet breakup process and shorter breakup lengths are obtained by increasing the inner and the outer gas velocity. Axially moving inner gas stream is more effective in disintegrating the annular sheet compared to axially moving outer gas stream. When both gas streams are moving axially, the liquid sheet breakup is quicker compared to that with any one gas stream. In the absence of outer gas swirl, the axisymmetric mode is the dominant instability mode. However, when outer gas flow has a swirl component higher helical modes become dominant. With increasing outer gas swirl strength, the maximum disturbance growth rate increases and the most unstable circumferential wave number increases resulting in a highly asymmetric sheet breakup with shorter breakup lengths and thinner ligaments.     

Instability of a Charged Droplet in an Inhomogeneous Electrostatic Field of a Rod of Finite Thickness

The instability of a charged droplet of an ideal liquid in an inhomogeneous electrostatic field of a rod of finite thickness maintained at a constant electrostatic potential is investigated within the framework of analytic asymptotic calculations. It is shown that the mode amplitudes and the drop oscillation frequencies increase with the rod thickness. The critical conditions of instability of the droplet reduce by several times as compared with the critical conditions of implementation of its instability in the electrostatic field of an infinitely thin filament maintained at a constant electrostatic potential. An analytic dependence between the charge and field parameters, critical for implementation of the instability of a charged droplet in an inhomogeneous electrostatic field and dependent on the rod thickness, is found.     

Axial instability of rotating rods revisited

For strain sufficiently small such that Hooke's Law is valid, it is shown that only a linear model for axial deformation of rotating rods can be derived. As discussed in the literature, this linear model exhibits an instability when the angular speed reaches a certain critical value. However, unless this linear model is valid for large strain, it is impossible to determine whether this instability really exists; because, as the angular speed is increased, the strain becomes large well short of the critical speed. Next, axial deformation of rotating rods is analyzed using two strain energy functions to model non-linear elastic behavior. The first of these functions is the usual quadratic strain energy function augmented with a cubic term. With this model it is shown that no instability exists if the non-linearity is stiffening (i.e. if the coefficient of the cubic term is positive), although the strain can become large. If the non-linearity is of the softening variety, then the critical angular speed drops as the degree of softening increases. Still, the strains are large enough that, except for rubber-like materials, a non-linear elastic model is not likely to be appropriate. The second strain energy function is based on the square of the logarithmic strain and yields a softening model. It quite accurately models the behavior of certain rubber rods which exhibit the instability within the validated range of elongation.     

Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power-law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.     

Computational study of the axial instability of rimming flow using Arnoldi method

Axial instability of rimming flow has been investigated by solving a linear generalized eigenvalue problem that governs the evolution of perturbations of two‐dimensional base flow. Using the Galerkin finite element method, full Navier–Stokes equations were solved to calculate base flow and this base flow was perturbed with three‐dimensional disturbances. The generalized eigenproblem formulated from these equations was solved by the implicitly restarted Arnoldi method using shift‐invert technique. This study presents instability curves to identify the critical wavelength of the neutral mode and the critical β, which measures the importance of gravity relative to viscosity. The axial instability of rimming flow is examined and three‐dimensional flow was reconstructed by using eigenvector and growth rate at a critical wave number. The critical β value in the axial instability analysis was observed to be comparable to the onset β value of the transition between the bump and the homogeneous film state in 2‐D base flow calculations. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonlinear shear-layer instability waves

The effects of critical-layer nonlinearity on spatially growing instability waves on shear layers between parallel streams are discussed. In the two-dimensional incompressible case, the flow in the critical layer is governed by a nonequilibrium (unsteady) nonlinear vorticity equation. The initial exponential growth of the instability wave is converted into algebraic growth during the streamwise aging of the critical layer into a quasi-equilibrium state. A uniformly valid composite formula for the instability wave amplitude, accounting for both nonparallel and nonlinear effects, is shown to be in good agreement with available experimental results. Nonlinear effects occur at smaller amplitudes for the three-dimensional and supersonic cases than in the two-dimensional incompressible case. The instability-wave amplitude evolution is then described by one integro-differential equation with a cubic-type nonlinearity, whose inviscid solution always end in a singularity at finite downstream distance.The US Government has the right to retain a nonexclusive royalty-free license in and to any copyright covering this paper.     

The nonlinear interaction of smooth travelling waves in an elastic string

It is shown that the collision of approaching weak travelling waves in a nonlinearly elastic leads to shock formation in finite time. Whereas the travelling waves are transverse, the shock is formed in the longitudinal waves generated by the collision.     

An analysis of unsteady highly turbulent swirling flow in a model vortex combustor

This paper reports an experimental investigation of a non-reacting turbulent swirling flow in a practical vortex combustor. The flow was examined for the conditions characteristic of the presence of a breakdown zone and a strong flow instability appearing at swirl numbers S>0.5. Flow visualization techniques, LDA measurements and acoustic probes were employed to study the unsteady flow characteristics. Based on the experimental results a positive first helical mode of instability was identified with a wavelength and frequency depending on swirl. The wavelength was confirmed to grow monotonically with S, while the dominant frequency of the flow pulsations was found to have an unusual parabolic evolution with swirl, with a minimum at S _min=0.88. This finding was interpreted using a proposed kinematic model based on the contribution of two mechanisms: rotation and axial motion of the helical vortex. It was concluded that for S<S _min the instability frequency is essentially dominated by the axial translation of the spiral vortex being inversely proportional to S and therefore giving a decreasing trend. For S>S _min the frequency of the flow precession is more dependent on the angular transportation of the vortex core, which resulted in the expected growing dependence on S.     

Stability of fluid flow in the presence of a compliant surface

This paper is concerned with the dynamic behaviour of a thin elastic plate in the presence of flow. The plate is excited by a time varying force applied on a line which is at right angles to the direction of flow. Causality is invoked to make the solution unique. It is found that the system responds differently for flow speeds above and below a critical value. Above this value, a temporal instability occurs and the amplitude of the motion grows with time at each point on the plate. Below this value there are travelling waves upstream of the driver, whilst downstream the behaviour is dominated by a disturbance which does not grow with time at any one position, but nevertheless grows as it is convected downstream. This type of instability is less severe than the temperal one and is of the type generally referred to as convective.     

Active Control of Wave Instabilities in Three-Dimensional Compressible Flows

In many fluid flows of practical importance transition is caused by the linear growth of wave instabilities, such as Tollmien–Schlichting waves, which eventually grow to a finite size at which stage secondary instabilities come into play. If transition is to be delayed or even avoided in such flows, then the linear growth of the disturbances must be prevented since control in the nonlinear regime would be a considerably more difficult task. Here a strategy for active control of two-dimensional incompressible and compressible Tollmien–Schlichting waves and its use in controlling the more practically relevant problem of crossflow instability which arises in swept-wing flows is discussed. The control is through an active suction/blowing distribution at the wall though the same result could be achieved by variable wall heating. In order to control the instability it is assumed that the wall shear stress and pressure are known from measurements. It is shown that, certainly at finite Reynolds numbers, it is sufficient to know the flow properties at a finite number of points along the wall. The cases of high and finite Reynolds numbers are discussed using asymptotic and numerical methods respectively. It is shown that a control strategy can be developed to stop the growth of all two-dimensional Tollmien–Schlichting waves at finite and large Reynolds numbers. Some discussion of nonlinear effects in the presence of active control is given and the possible control of other instability mechanisms investigated. Received 1 May 1998 and accepted 24 September 1998     

Inviscid evolution of incompressible swirling flows in pipes: The dependence of the flow structure upon the inlet velocity field

This paper analyses the influence of the inlet swirl on the structure of incompressible inviscid flows in pipes. To that end, the inviscid evolution along a pipe of varying radius with a central body situated inside the pipe is studied for three different inlet swirling flows by solving the Bragg–Hawthorne equation both asymptotically and numerically. The downstream structure of the flow changes abruptly above certain threshold values of the swirl parameter (L). In particular, there exist a value L_r above which a near-wall region of flow reversal is formed downstream, and a critical value L_f above which the axial vortex flow breaks down. It is shown that the dependence upon the pipe geometry of these critical values of the swirl parameter varies strongly with the inlet azimuthal velocity profile considered. An excellent agreement between asymptotic and numerical results is found.