Dynamic Stability of a Fluid-Coveying Cantilevered Pipe on an Additional Combined Support

Based on an analytical study, a numerical analysis is made of the dynamic stability of a cantilevered steel pipe conveying a fluid. The pipe is modeled by a beam restrained at the left end and supported by a special device (a rotational elastic restraint plus a Q-apparatus) at the right end. The numerical analysis reveals that the critical velocity of the fluid depends on the governing parameters of the problem such as the ratio of the fluid mass to the pipe mass per unit length and the rotational elastic constant at the right end     

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.     

Reflection and radiation of a wave system at the open end of a submerged elastic pipe

The reflection and radiation of a wave system at the open end of a submerged semi-infinite elastic pipe are studied. This wave system consists of a flexural wave in the pipe, an acoustic surface wave in the fluid exterior to the pipe and an acoustic wave in the pipe’s interior. Fourier transform techniques are used to formulate this semi-infinite geometry problem rigorously as a Wiener-Hopf type equation. An approximate solution is obtained by using a perturbation method in which the ratio of the massdensities of the fluid and the pipe material is regarded as a small parameter. The calculation of the reflection coefficient is emphasized, and the polar plots of the radiation coefficient are also presented.     

基于三阶剪切变形理论的悬臂输流管道自由振动

采用三阶剪切变形理论,结合有限元法研究了悬臂输流管道的自由振动问题．利用虚功原理建立了输流管系统的有限元方程,同时将悬臂端弹性支承以势能的形式引入到系统方程中,求解了系统前三阶的复频率．分别探讨了流体速度和弹簧刚度对系统复频率实部和虚部的影响,重点分析了弹簧刚度与前三阶固有频率间的关系．在弹性支承刚度为零的特例下,对比了本文结果与Timoshenko梁理论的结果,证明了本文方法的可靠性．研究发现系统固有频率的实部恒为负值,表明一端带有弹性支承的约束形式有利于提高悬臂输流管道自由振动的稳定性;流体的流动对管道振动起到了阻尼作用,在流动速度足够大的情况下,各阶振动固有频率均趋于零;当弹簧刚度为无穷大,且流体速度足够大时,输流管道将发生失稳．     

Experimental investigation of the fluid–structure interaction in an elastic 180° curved vessel at laminar oscillating flow

Fluid–structure interaction phenomena are extremely important when laminar flows through elastic vessels such as in biomedical flow problems are considered. In general, such elastic vessels are curved which is why an elastic 180° bend at a curvature ratio \(\delta = D/D_{\rm C} = 0.\bar{2}\) defines the reference geometry in this study. It is the purpose of this study to compare the results with the steady flow through a 180° rigid pipe bend and to quantify the impact of the fluid–structure interaction on the overall flow pattern and the vessel deformation at oscillating fully developed entrance flow. The findings comprise velocity, pressure, and structure deformation measurements. The vessel dilatation amplitude was varied between 3.75 % and 7 % of the vessel diameter at Dean De and Womersley number Wo ranges of \(327\,\le\,De\,\le\,350\) and \(7\,\le\,Wo\,\le\,8.\) The flow is investigated by time-resolved stereoscopic particle-image velocimetry in five radial cross sections located in the elastic 180° bend and in the inlet pipes. The unsteady static vessel pressure is measured synchronously at these cross sections. The comparison of the steady with the unsteady flow field shows a strong change in the axial and secondary velocity distributions at periods of transition between the centrifugal forces and the unsteady inertia forces dominated regimes. These changes are characterized by asymmetric fluctuations of the centers of the counter-rotating vortex pair. The investigation of the impact of the structure deformation amplitude on these fluctuations reveals a significant attenuation at high deformation amplitudes. The spatial motion of the elastic vessel due to the forces applied by the flow exhibits amplitudes up to 15 % of the vessel diameter. Considering the fluid–structure interaction, an amplification of the volume flux amplitude by a factor of 2.1 at the vessel outlet and phase lags up to 30° occur. The static pressure distribution is characterized by a pronounced asymmetry between forward and backward flow with a 40 % higher peak magnitude at backward flow and phase lags of 35°. The results evidence that a strong distortion of the velocity distribution in the bend, which is caused by the oscillating nature of the flow, is reduced as a result of the fluid–structure interaction.     

输液管道附加支承刚度优化设计

研究液固耦合效应作用下,两端铰支输液管道系统附加支承的刚度和位置优化设计。应用有限元分析方法,建立了输液管道液固耦合振动方程。为有效控制管道结构的振动,利用在管道结构上附(增)加支承的方法,提高输液管道系统的固有频率,预防系统可能发生强烈的耦合振动导致不稳定状态。提出了附加支承最小(临界)刚度的快速计算策略和途径,分别探讨分析了输液管道内液体的流速、附加支承的位置以及第一阶固有频率的目标值对最优支承刚度值的影响。     

Numerical analysis of the buckling and recuperation dynamics of flexible filament using an immersed boundary framework

The dynamics of flexible filaments in viscous shear flow is of interest to biologists and engineers in a wide variety of applications involving folding and unfolding sequence of long-chain biomolecules like DNA, non-motile sperm and microalgae. It is also helpful in understanding the deformation of natural and synthetic fibers which can be applied in areas such as biotechnology. In the present work, deformation and migration behavior of non-motile unicellular phytoplankton diatoms subjected to viscous shear flow are considered. These unicellular diatoms develop into colonies which are made up of linked chains. The complex fluid-structure interaction is solved by developing a two-dimensional numerical model with an immersed boundary framework. The simulation consists of suspending an elastic filament mimicking a diatom chain in a shear flow at low Reynolds number. The governing continuity and Navier–Stokes equations are solved on a Cartesian grid arranged in a staggered manner. A forcing term is added to the momentum equation that incorporates the presence of flexible filament in the fluid domain. The discretization of the governing equation is based on a finite volume method, and a SIMPLE algorithm is used to compute pressure and velocity. A computer code is developed to perform numerical simulations, and the model is first verified with the deformation study of a tethered flexible filament in uniform fluid flow. Next, the shape deformations for flexible filament placed freely in shear flow are compared with the studies of previous researchers. Further, the present results are validated with Jeffery's equation for particles immersed in shear flow along with classification plot for filament orbit regimes. All of these comparisons provide a reasonable validity for the developed model. The effect of bending rigidity and shear rate on the deformation and migration characteristics is ascertained with the help of parametric studies. A non-dimensional parameter called Viscous Flow Forcing value (VFF) is calculated to quantify the parametric results. An optimum Viscous Flow Forcing value is determined which indicates the transition of filaments exhibiting either a recuperative (regaining original shape past deformation) or non-recuperative (permanently deformed) behavior. The developed model is successful in capturing fluid motion, diatom buckling, shape recurrences and recuperation dynamics of diatom chains subjected to shear flow. Further, the developed computational model can successfully illustrate filament-fluid interaction for a wide variety of similar problems.     

Analysis of coupled-mode flutter of pipes conveying fluid on the elastic foundation

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两端弹性支承输流管道固有特性研究

输流管道广泛应用于航天航空、石油化工、海洋等重要的工程领域, 其振动特性尤其是系统固有特性一直是国内外学者研究的热点问题. 本文研究了两端弹性支承输流管道横向振动的固有特性, 尤其是在非对称弹性支承下的系统固有特性. 使用哈密顿原理得到了输流管道的控制方程及边界条件, 通过复模态法得到了静态管道的模态函数, 以其作为伽辽金法的势函数和权函数对线性派生系统控制方程进行截断处理. 分析了两端对称支承刚度、两端非对称支承刚度、管道长度以及流体质量比对系统固有频率的影响规律, 重点讨论了管道两端可能形成的非对称支承条件下固有频率的变化规律. 结果表明, 较大的对称支承刚度下管道的第一阶固有频率下降较快; 当管道两端支承刚度变化时, 管道的各阶固有频率在两端支承刚度相等时取得最值; 对于两端非对称支承的管道而言, 两端支承刚度越接近, 第一阶固有频率下降的越快, 而且相应的临界流速越小; 流体的流速越大, 其对两端非对称弹簧支承的管道固有频率的影响更为明显.      

Experimental investigation of viscoelastic drop deformation in Newtonian matrix at high capillary number under simple shear flow

The steady state deformation of a viscoelastic drop (Boger fluid) in a Newtonian liquid at high capillary number under simple shear flow is investigated by direct visualization using a specially designed Couette apparatus which enables visualization from two perpendicular directions. Two drop deformation modes are found: (1) Mode I – drop deformation in the flow direction and (2) Mode II – drop deformation in the vorticity direction. The drop deformation mode depends on the relative strength of the elastic contribution to viscous contribution. If the elastic contribution is weak compared to the viscous contribution, the drop elongates in the flow direction via Mode I. If the elastic contribution is strong, the drop elongates in the vorticity direction via Mode II. The drop size also affects the drop deformation. At the same capillary number, bigger drops have larger deformations than smaller drops.     

Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite element method

Fluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self-excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non-linear equations governing even the most simplified two-dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time-dependent problem might be approached.     

Divergent Instability Domains of a Fluid-Conveying Cantilevered Pipe with a Combined Support

The divergent instability of a fluid-conveying pipe is analyzed numerically. The pipe is modeled by a cantilevered beam restrained at the left end and supported by a special device (a rotational elastic restraint plus a Q-apparatus) at the right end. Divergent instability domains of the pipe are obtained by varying the rotational elastic constant of the right restraint     

Wave propagation in a fluid flowing through a curved thin-walled elastic tube

The pulsatile flow in a curved elastic pipe of circular cross section is investigated. The unsteady flow of a viscous fluid and the wall motion equations are written in a toroidal coordinate system, superimposed and linearized over a steady state solution. Being the main application relative to the vascular system, the radius of the pipe is assumed small compared with the radius of curvature. This allows an asymptotic analysis over the curvature parameter. The model results an extension of the Womersley's model for the straight elastic tube. A numerical solution is found for the first order approximation and computational results are finally presented, demonstrating the role of curvature in the wave propagation and in the development of a secondary flow.     

三参量固体模型粘弹性输流管道的动力特性分析

推导了三参量固体模型粘弹性输流管道的振动微分方程 ,计算了在不同无量纲松弛系数和弹性常数比下管道的无量纲临界流速和无量纲自振复频率 ,并给出了前三阶复频率与流速的关系 .计算结果表明 ,质量比、无量纲松弛系数及无量纲弹性常数比对输流管道的动力特性均有影响 .     

A perturbation solution for non-linear vibration of uniformly curved pipes conveying fluid

The first-order non-linear interactions between the pipe structure and the flowing fluid are considered to formulate the governing equations of motion for the in-plane vibration of a circular-arc pipe containing flowing fluid. The forces and moments induced in a pipe element by the flowing fluid are analyzed as functions of the instantaneous local curvature of the pipe. The flow field is assumed to be one-dimensional, incompressible and of uniform flow, and to remain independent of pipe motion. For a fixed-end circular-arc pipe with arbitrary arc angle, the non-linear governing equations are solved by the method of multiple scales in conjunction with the Bubnov-Galerkin method. The non-linear solutions indicate that the vibrational behavior of the system can differ substantially from that predicted by a linear analysis.     

Dynamic coupling of fluid and structural mechanics for simulating particle motion and interaction in high speed compressible gas particle laden flow

In this work, structural finite element analyses of particles moving and interacting within high speed compressible flow are directly coupled to computational fluid dynamics and heat transfer analyses to provide more detailed and improved simulations of particle laden flow under these operating conditions. For a given solid material model, stresses and displacements throughout the solid body are determined with the particle–particle contact following an element to element local spring force model and local fluid induced forces directly calculated from the finite volume flow solution. Plasticity and particle deformation common in such a flow regime can be incorporated in a more rigorous manner than typical discrete element models where structural conditions are not directly modeled. Using the developed techniques, simulations of normal collisions between two 1 mm radius particles with initial particle velocities of 50–150 m/s are conducted with different levels of pressure driven gas flow moving normal to the initial particle motion for elastic and elastic–plastic with strain hardening based solid material models. In this manner, the relationships between the collision velocity, the material behavior models, and the fluid flow and the particle motion and deformation can be investigated. The elastic–plastic material behavior results in post collision velocities 16–50% of their pre-collision values while the elastic-based particle collisions nearly regained their initial velocity upon rebound. The elastic–plastic material models produce contact forces less than half of those for elastic collisions, longer contact times, and greater particle deformation. Fluid flow forces affect the particle motion even at high collision speeds regardless of the solid material behavior model. With the elastic models, the collision force varied little with the strength of the gas flow driver. For the elastic–plastic models, the larger particle deformation and the resulting increasingly asymmetric loading lead to growing differences in the collision force magnitudes and directions as the gas flow strength increased. The coupled finite volume flow and finite element structural analyses provide a capability to capture the interdependencies between the interaction of the particles, the particle deformation, the fluid flow and the particle motion.     

A lattice Boltzmann fictitious domain method for modeling red blood cell deformation and multiple‐cell hydrodynamic interactions in flow

To model red blood cell (RBC) deformation and multiple‐cell interactions in flow, the recently developed technique derived from the lattice Boltzmann method and the distributed Lagrange multiplier/fictitious domain method is extended to employ the mesoscopic network model for simulations of RBCs in flow. The flow is simulated by the lattice Boltzmann method with an external force, while the network model is used for modeling RBC deformation. The fluid–RBC interactions are enforced by the Lagrange multiplier. To validate parameters of the RBC network model, stretching tests on both coarse and fine meshes are performed and compared with the corresponding experimental data. Furthermore, RBC deformation in pipe and shear flows is simulated, revealing the capacity of the current method for modeling RBC deformation in various flows. Moreover, hydrodynamic interactions between two RBCs are studied in pipe flow. Numerical results illustrate that the leading cell always has a larger flow velocity and deformation, while the following cells move slower and deform less.Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of viscous flow interaction with an elastic membrane

A numerical fluid–structure interaction model is developed for the analysis of viscous flow over elastic membrane structures. The Navier–Stokes equations are discretized on a moving body‐fitted unstructured triangular grid using the finite volume method, taking into account grid non‐orthogonality, and implementing the SIMPLE algorithm for pressure solution, power law implicit differencing and Rhie–Chow explicit mass flux interpolations. The membrane is discretized as a set of links that coincide with a subset of the fluid mesh edges. A new model is introduced to distribute local and global elastic effects to aid stability of the structure model and damping effects are also included. A pseudo‐structural approach using a balance of mesh edge spring tensions and cell internal pressures controls the motion of fluid mesh nodes based on the displacements of the membrane. Following initial validation, the model is applied to the case of a two‐dimensional membrane pinned at both ends at an angle of attack of 4° to the oncoming flow, at a Reynolds number based on the chord length of 4 × 10³. A series of tests on membranes of different elastic stiffness investigates their unsteady movements over time. The membranes of higher elastic stiffness adopt a stable equilibrium shape, while the membrane of lowest elastic stiffness demonstrates unstable interactions between its inflated shape and the resulting unsteady wake. These unstable effects are shown to be significantly magnified by the flexible nature of the membrane compared with a rigid surface of the same average shape. Copyright © 2007 John Wiley & Sons, Ltd.     

充液矩形贮箱弹性箱底板应力与变形分析

基于相容拉格朗日-欧拉法,通过对流场与弹性固体间流固耦合作用的分析,得到了矩形贮箱弹性底板流固耦合系统的自由振动微分方程。将伯努利方程与外加激励条件、速度势函数耦合到自由振动方程中,采用迦辽金积分法,给出了矩形贮箱在流体作用下的应力与变形的解析解。讨论了弹性底板的抗弯刚度、结构尺寸、底板材料参数及流体深度等因素对底板应力与变形的影响。研究结果表明:在液体晃动非线性激励作用下,贮箱底板的应力和变形随着液体深度、板长的增大而增大,随着板厚的减小而增大,且成非线性变化关系;底板的变形及应力与底板的材料常数相关,其中板厚的变化对其变形和应力影响要比板长及液体深度的影响显著得多。本文结果可为工程实际中矩形贮箱的设计提供参考。     

Oscillating laminar fluid flow in a cylindrical elastic pipe of annular cross-section

The coupled elastohydrodynamic problem based on the dynamic equations for a viscous incompressible fluid and for two closed finite-length cylindrical elastic shells, inner and outer, described using the Kirchhoff-Love hypotheses is formulated and solved with the corresponding boundary conditions for harmonic variation of the pressure at the inlet and outlet of an elastic annular pipe. From the solution of this problem the flow parameters and the elastic shell displacements are found. The amplitude and phase frequency characteristics and resonant frequencies of the shells are found. The cases of shells simply supported and with fixed ends are considered. The effect of the support mode and the fluid characteristics on the resonant frequencies and the amplitude frequency characteristics of the shells is investigated.