Dynamics of a long tubular cantilever conveying fluid downwards,which then flows upwards around the cantilever as a confined annular flow

A theoretical model is developed for the dynamics of a hanging tubular cantilever conveying fluid downwards; the fluid, after exiting from the free end, is pushed upwards in the outer annular region contained by the cantilever and a rigid cylindrical channel. This configuration thus resembles that of a drill-string with a floating fluid-powered drill-bit. The linear equation of motion is solved by means of a hybrid Galerkin–Fourier method, as well as by a conventional Galerkin method. Calculations are conducted for a very slender system with parameters appropriate for a drill-string, for different degrees of confinement of the outer annular channel; and also for another, bench-top-size experiment. For wide annuli, the dynamics is dominated by the internal flow and, for low flow velocities, the flow increases the damping associated with the presence of the annular fluid. For narrow annuli, however, the annular flow is dominant, tending to destabilize the system, giving rise to flutter at remarkably low flow velocities. The mechanisms underlying the dynamics are also considered, in terms of energy transfer from the fluid to the cantilever and vice versa, as are possible applications of this work.     

环形截面螺旋管道内二次流动特性的研究

从曲线柱坐标系下的N-S方程出发,以曲率和挠率为小参数,采用摄动法求解了环形截面螺旋管道内的黏性流动,给出了完全二阶摄动解,结果表明：当挠率为零时,二次流表现为上下对称的四个涡;当挠率不为零,涡的对称性遭到破坏,二次涡的强度和个数受De数和环形截面内外径之比δ的影响,轴向速度最大值在De数较小时靠近管道的内侧,随着De数的增加,其最大值向外侧移动。     

Dynamics of a partially confined,discharging, cantilever pipe with reverse external flow

The potential for fluid-elastic instability of hanging cantilevered pipes subjected to simultaneous internal and external axial flows is investigated. Such systems may lose stability by amplified oscillations (flutter) or buckling (static divergence). The system of interest is a flexible tubular cantilever hanging concentrically within a rigid outer tube of larger diameter. Flow inside the cantilever is directed from the clamped end to the free end. Upon exiting the cantilever, the fluid flows in the opposite direction in the annular region between the outer tube and the cantilever. The rigid outer tube is of variable length and it can cover part of the length of the cantilever. This system has applications in brine production and salt-cavern hydrocarbon storage. A linear model is derived based on the work of Paidoussis, Luu and Prabhakar; the presence of the shorter outer rigid tube is taken into account in a simplified way. Series solutions are obtained using a Galerkin method with Euler–Bernoulli beam eigenfunctions as comparison functions. Experimental results are presented and compared with the theoretical model. Additional computations are performed to quantify the effect of confinement (i.e. the narrowness of the annular region) on the cantilever stability, as well as the effect of confined-flow length, for both the short laboratory-sized system and long brine-string-like systems. An increase in these parameters gives rise to flutter for short systems, or a succession of flutter and divergence for long systems. In addition, the effect of the system length is investigated. Increasing length results in asymptotic behaviour, with both the critical flow-velocity and associated frequency reaching limiting values. Sufficiently long systems lose stability by divergence rather than flutter.     

DIVERGENCE INSTABILITY OF VARIABLE-ARC-LENGTH ELASTICA PIPES TRANSPORTING FLUID

A flexible elastic pipe transporting fluid is held by an elastic rotational spring at one end, while at the other end, a portion of the pipe may slide on a frictional support. Regardless of the gravity loads, when the internal flow velocity is higher than the critical velocity, large displacements of static equilibrium and divergence instability can be induced. This problem is highly nonlinear. Based on the inextensible elastica theory, it is solved herein via the use of elliptic integrals and the shooting method. Unlike buckling with stable branching of a simply supported elastica pipe with constant length, the variable arc-length elastica pipe buckles with unstable branching. The friction at the support has an influence in shifting the critical locus over the branching point. Alteration of the flow history causes jumping between equilibrium paths due to abrupt changes of direction of the support friction. The elastic rotational restraint brings about unsymmetrical bending configurations; consequently, snap-throughs and snap-backs can occur on odd and even buckling modes, respectively. From the theoretical point of view, the equilibrium configurations could be formed like soliton loops due to snapping instability.     

复杂载荷下双金属复合管的屈曲失效研究

本文采用有限元方法系统地研究了复杂载荷下双金属复合管的屈曲失效，三维有限元数值模型考虑了双金属复合管的准静态复合成型制造过程中产生的残余应力，分析了外基管直径、内衬管壁厚、内外管初始间隙、内衬管屈服强度、内压等因素对双金属复合管屈曲失效的影响。结果表明，加载路径、复合管的几何尺寸及内衬管的屈服强度对双金属复合管的屈曲性能均有较大影响，内充压力可以延迟内衬管的屈曲失效。     

MHD liquid metal flow and heat transfer between vertical coaxial cylinders under horizontal magnetic field

Direct numerical simulations are presented of MHD liquid metal flow and heat transfer in vertical annuli. Three annular gaps and four ratios of annular height to annular gap are considered. The walls of the external and internal cylinders are isothermal with the temperature of the outer cylinder being higher and, thus, buoyancy is the driving force. The results show that the fluid motion increases as the aspect ratio and the annular gap become larger. The presence of the magnetic field results to fluid deceleration and, thus, to flow stabilization. Additionally, non symmetric flow patterns develop, due to the magnetic field, resulting in differently sized normal and parallel wall layers, namely the Hartmann and the Roberts layers, respectively. For all annular gaps considered, the highest spatially averaged heat transfer rates are obtained for aspect ratios equal to 1.     

Mechanical behavior and wrinkling of lined pipes

The paper focuses on wrinkling of lined pipes (sometimes referred to as clad pipes) under bending loading, where a corrosion-resistant thin-walled liner is fitted inside a carbon–steel outer pipe. The problem is solved numerically, using nonlinear finite elements to simulate liner pipe deformation and its interaction with the outer pipe. Stresses and strains are monitored throughout the deformation stage, detecting possible detachment of the liner from the outer pipe and the formation of wrinkles. The wrinkling behavior of elastic and elastic–plastic (steel) lined pipes under bending is examined. The results indicate that the lateral confinement of the liner pipe due to the deformable outer pipe and its interaction with the outer pipe has a decisive influence on the wrinkling behavior of the lined pipe. It is also shown that the behavior is characterized by a first bifurcation in a uniform wrinkling pattern, followed by a secondary bifurcation. The values of corresponding buckling curvature are determined and comparison with available experimental results is conducted in terms of wrinkle height development and the corresponding buckling wavelength. The results of the present research can be used for safer design of lined pipes in pipeline applications.     

Hydroelastic vibration of two annular plates coupled with a bounded compressible fluid

A theoretical study on a linear hydroelastic vibration of two annular plates coupled with a bounded fluid is presented. The proposed method, based on the Rayleigh–Ritz method and the finite Hankel transform, is verified through a finite element analysis by using a commercial computer code, with an excellent accuracy. It is assumed that plates with an unequal thickness and with an unequal inner radius are clamped along their edges and an inviscid compressible fluid fills the space between the annular plates and the outer rigid vessel. When the two annular plates are identical, distinct in-phase and out-of-phase modes are observed. By increasing the difference in the plate thickness, the symmetric in-phase and out-of-phase modes with respect to the middle plane of the system are gradually shifted to pseudo in-phase and out-of-phase modes, and eventually they are changed to mixed modes. It is found that the natural frequencies decrease with an increase of the fluid compressibility, and additional modes due to a fluid concentration are observed when the plates are coupled with a compressible fluid. The fluid compressibility effect on the natural frequency is dominant in the out-of-phase modes and the higher modes. Also, the effects of the fluid thickness or the distance between the plates and the inner radius of the plates on the natural frequencies of the wet modes are investigated.     

研究流体在偏心环空内流动的新方法

使用双极坐标系研究流体在偏心环空内的流动。给出该坐标系下流体力学基本方程组。对牛顿流体的轴向流、旋转流求得它们的精确级数解和相应的数值结果。     

Numerical investigation of fully developed laminar flow in irregular annuli: Square‐circular ducts

Numerical study is carried out on fully developed laminar axial flow of incompressible Newtonian fluid in irregular annuli of different cross‐sectional geometry. Accurate prediction of the friction factor of this flow was obtained using high‐order finite element method. Investigations are performed for the annular flow between rectangular and circular ducts under all possible arrangements. The governing equations are solved by an efficient finite element technique. It was found that a higher annular area‐ratio will lead to a monotonic increase in (f Re) value in the case of regular annuli, and will lead to an increase followed by a decrease in (f Re) value in the case of irregular annuli. Furthermore, this study illustrates that irregular annuli have lower (f Re) value than regular annuli, and that the square‐in‐circle case (SC‐case) has the lowest (f Re) value whereas the circle‐in‐circle case (CC‐case) has the highest (f Re) value. The finite element technique developed in this study was validated against the already reported numerical results in the literature, and the results presented in this work show excellent agreement. Copyright © 2009 John Wiley & Sons, Ltd.     

圆筒内旋转细长管与流体界面耦合的数值方法

圆筒内旋转细长管是石油钻采工程中特有结构,细长管不仅与圆筒发生碰撞接触,还与管内流体和管外环空流体耦合,是一个复杂的非线性流固耦合系统。细长管固体域离散成梁单元,采用非线性碰撞接触动力学方程描述;管内外流体离散成六面体单元,采用计算流体动力学方程描述,在耦合界面处用任意拉格朗日欧拉法动网格来处理运动界面。根据梁单元位移...     

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

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Investigation of a combustion driven oscillation in a refinery flare: Part B: Visualisation of a periodic flow instability in a bifurcating duct following a contraction

A flow visualisation study was performed to investigate a periodic flow instability in a bifurcating duct within the tip of the flares at the Shell refinery in Clyde, NSW, to verify the trigger of a combustion-driven oscillation proposed in Part A of this study, and to identify its features. The model study assessed only the flow instability, uncoupled from the acoustic resonance and the combustion that are also present in the actual flare. Three strong, coupled flow oscillations were found to be present in three regions of the fuel line in the flare tip model. A periodic flow separation was found to occur within the contraction at the inlet to the tip, a coupled, periodic flow oscillation was found in the two transverse “cross-over ducts” from the central pipe to the outer annulus and an oscillating flow recirculation was present in the “end-cap” region of the central pipe. The dimensionless frequency of these oscillations in the model was found to match that measured in the full-scale plant for high fuel flow rates. This, and the strength of these flow oscillations, gives confidence that they are integral to the full-scale combustion-driven oscillation and most likely the primary trigger. The evidence indicates that the periodic flow instability is initiated by the separation and roll-up of the annular boundary layer at the start of the contraction in the fuel section of the flare tip. The separation generates an annular vortex which interacts with the blind-ended pipe downstream, leading to a pressure wave which propagates back upstream, initiating the next separation event and repeating the cycle. The study also investigated flow control devices with a view to finding a practical approach to mitigate the oscillations. The shape of these devices was constrained to allow installation without removing the tip of the flare. This aspect of the study highlighted the strength and nature of the coupled oscillation, since it proved to be very difficult to mitigate the oscillation in this way. An effective configuration is presented, comprising of three individual components, all three of which were found to be necessary to eliminate the oscillation completely.     

Nonlinear dynamics of functionally graded pipes conveying hot fluid

For improved stability of fluid-conveying pipes operating under the thermal environment, functionally graded materials (FGMs) are recommended in a few recent studies. Besides this advantage, the nonlinear dynamics of fluid-conveying FG pipes is an important concern for their engineering applications. The present study is carried out in this direction, where the nonlinear dynamics of a vertical FG pipe conveying hot fluid is studied thoroughly. The FG pipe is considered with pinned ends while the internal hot fluid flows with the steady or pulsatile flow velocity. Based on the Euler–Bernoulli beam theory and the plug-flow model, the nonlinear governing equation of motion of the fluid-conveying FG pipe is derived in the form of the nonlinear integro-partial-differential equation that is subsequently reduced as the nonlinear temporal differential equation using Galerkin method. The solutions in the time or frequency domain are obtained by implementing the adaptive Runge–Kutta method or harmonic balance method. First, the divergence characteristics of the FG pipe are investigated and it is found that buckling of the FG pipe arises mainly because of temperature of the internal fluid. Next, the dynamic characteristics of the FG pipe corresponding to its pre- and post-buckled equilibrium states are studied. In the pre-buckled equilibrium state, higher-order parametric resonances are observed in addition to the principal primary and secondary parametric resonances, and thus the usual shape of the parametric instability region deviates. However, in the post-buckled equilibrium state of the FG pipe, its chaotic oscillations may arise through the intermittent transition route, cyclic-fold bifurcation, period-doubling bifurcation and subcritical bifurcation. The overall study reveals complex dynamics of the FG pipe with respect to some system parameters like temperature of fluid, material properties of FGM and fluid flow velocity.     

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

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

Entropy analysis for third grade fluid flow with Vogel model viscosity in annular pipe

The steady-state flow of a third grade fluid between concentric circular cylinders is considered and entropy generation due to fluid friction and heat transfer in the annular pipe is examined. Depending upon the fluid viscosity, entropy generation in the flow system varies. The third grade fluid is employed to account for the non-Newtonian effect while Vogel model is accommodated for temperature-dependent viscosity. The analysis is based on perturbation technique. The closed form solutions for velocity, temperature and entropy fields are presented. Entropy generation due to fluid friction and heat transfer in the flow system is formulated. The influence of viscosity parameters A and B on the entropy generation number is investigated. It is found that entropy generation number reduces with increasing viscosity parameter A, which is more pronounced in the region close to the annular pipe inner wall and opposite is true for increasing viscosity parameter B.     

INSTABILITY AND CHAOS IN A PIPE CONVEYING FLUID WITH ADDED MASS AT FREE END

This paper shows the mechanism of instability and chaos in a cantilevered pipe conveying steady fluid. The pipe under consideration has added mass or a nozzle at the free end. The Galerkin method is used to transform the original system into a set of ordinary differential equations and the standard methods of analysis of the discrete system are introduced to deal with the instability. With either the nozzle parameter or the flow velocity increasing, a route to chaos can be observed very clearly: the pipe undergoing buckling (pitchfork bifurcation), flutter (Hopf bifurcation), doubling periodic motion (pitchfork bifurcation) and chaotic motion occurring finally. The project supported by the National Key Projects of China under grant No. PD9521907 and Science Foundation of Tongji University under grant No. 1300104010.     

Dynamics of interaction modes in excited annular jets

Evolution of coherent structures and their interaction dynamics are educed in the near field of an acoustically excited basic annular jet using conditional sampling technique based on a multiple triggering criterion to detect the two dominating modes of structure pattern. Acoustic excitation is applied with an aim to better organize the phase alignment of initial rolling and pairing process in the outer shear layer. Negligible modification of the time-averaged flow field results from the excitation. The educed coherent vorticities show that the two modes of evolution are due to the corresponding two modes of shedding pattern of the wake structures from the centerbody, namely the mode one wake and the mode zero wake. In both modes, the shear-layer mode jet vortex rings in the outer layer are perturbed by the shedding of wake structures in the inner region and interaction involving primary merging of three successive jet vortex rings or their partial circumferential sections is found. This results in the formation of wake-induced structures of the corresponding mode pattern, which possesses concentration of coherent vorticity and fluid circulation over a large spatial extent at 1 < x/D < 2. Secondary interactions, such as vortex tearing, are also observed.     

Pseudo-nonlinear dynamic analysis of buckled pipes

In this study, the post-divergence behavior of fluid-conveying pipes supported at both ends is investigated using the nonlinear equations of motion. The governing equation exhibits a cubic nonlinearity arising from mid-plane stretching. Exact solutions for post-buckling configurations of pipes with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are investigated. The pipe is stable at its original static equilibrium position until the flow velocity becomes high enough to cause a supercritical pitchfork bifurcation, and the pipe loses stability by static divergence. In the supercritical fluid velocity regime, the equilibrium configuration becomes unstable and bifurcates into multiple equilibrium positions. To investigate the vibrations that occur in the vicinity of a buckled equilibrium position, the pseudo-nonlinear vibration problem around the first buckled configuration is solved precisely using a new solution procedure. By solving the resulting eigenvalue problem, the natural frequencies and the associated mode shapes of the pipe are calculated. The dynamic stability of the post-buckling configurations obtained in this manner is investigated. The first buckled shape is a stable equilibrium position for all boundary conditions. The buckled configurations beyond the first buckling mode are unstable equilibrium positions. The natural frequencies of the lowest vibration modes around each of the first two buckled configurations are presented. Effects of the system parameters on pipe behavior as well as the possibility of a subcritical pitchfork bifurcation are also investigated. The results show that many internal resonances might be activated among the vibration modes around the same or different buckled configurations.     

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.