流体黏性对输流管道运动方程及临界流速的影响

考虑实际流体黏性引起的管内流速非均匀分布,针对层流和两种不同的湍流流态,对理想流体情况下输流管道运动方程中的离心力项进行了修正,得到的修正系数分别为1.333(圆管层流)、1.020(光滑管壁圆管湍流)和1.037～1.055(粗糙管壁圆管湍流).根据修正后的运动方程得到的上述3种情况下的发散失稳临界流速比理想流体流动情况下依次分别低13.4%,1.0%和1.8%～2.6%.流体黏性对输流管道运动方程及临界流速的影响只与流态有关,雷诺数决定流态,而黏性系数通过雷诺数间接起作用.     

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.     

分布式运动约束下悬臂输液管的参数共振研究

输液管道结构在航空、航天、机械、海洋、水利和核电等工程领域都有广泛应用,其稳定性、振动与安全评估备受关注.针对具有分布式运动约束悬臂输液管的非线性动力学模型,分别采用立方非线性弹簧和修正三线性弹簧来模拟运动约束的作用力,研究了管道在脉动内流激励下的参数共振行为.首先,从输液管系统的非线性控制方程出发,利用Galerkin方法进行离散化;然后,由Floquet理论得出线性系统在失稳前两个不同平均流速下脉动幅值和脉动频率变化时的共振参数区域;最后,考虑系统的几何非线性项和分布式非线性运动约束力的影响,求解了管道的非线性动力学响应,讨论了非线性项及运动约束力对管道参数共振行为的影响.研究结果表明,系统非线性共振响应的参数区域与线性系统的共振参数区域是一致的,分布式运动约束力对发生参数共振时管道的位移响应有显著影响;立方非线性弹簧和修正三线性弹簧模型所预测的分岔路径存有较大差异,但都可诱发管道在一定的参数激励下出现混沌运动.      

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.     

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.     

In-plane dynamics of a fluid-conveying corrugated pipe supported at both ends

The dynamics and stability of fluid-conveying corrugated pipes are investigated. The flow velocity is assumed to harmonically vary along the pipe rather than with time. The dimensionless equation is discretized with the differential quadrature method(DQM). Subsequently, the effects of the mean flow velocity and two key parameters of the corrugated pipe, i.e., the amplitude of the corrugations and the total number of the corrugations, are studied. The results show that the corrugated pipe will lose stability by flutter even if it has been supported at both ends. When the total number of the corrugations is sufficient, this flutter instability occurs at a micro flow velocity. These phenomena are verified via the Runge-Kutta method. The critical flow velocity of divergence is analyzed in detail. Compared with uniform pipes, the critical velocity will be reduced due to the corrugations, thus accelerating the divergence instability. Specifically,the critical flow velocity decreases if the amplitude of the corrugations increases. However, the critical flow velocity cannot be monotonously reduced with the increase in the total number of the corrugations. An extreme point appears, which can be used to realize the parameter optimization of corrugated pipes in practical applications.     

Use of the equation for turbulent viscosity to describe the flow near a rough surface

An investigation of the flow at a rough surface, as well as in pipes and channels with rough walls, is one of the most important problems of applied hydrodynamics. Results of classical investigations, in which the most important flow properties near a rough surface are clarified, are generalized in [1–3]. These investigations are the basis for the construction of numerous semiempirical theories using the “mixing path∝ model of L. Prandtl (for instance, [4–6]). However, despite their simplicity these methods possess all the disadvantages inherent in the Prandtl theory: They are not universal, they describe the transition from the laminar to the turbulent mode poorly, and they are not applicable for the computation of complex non-self-similar flows. Meanwhile, an analysis of the experimental results obtained in [7], for example, indicates an extremely complex flow structure both in the neighborhood of the rough surface and far away from it. Models using the differential equation of the kinetic energy of turbulence have recently been developed to describe turbulent flow near a rough surface [8]. The possibilities of applying a model using the equation for turbulent viscosity to close the problem [9] are analyzed in this paper in an example of a steady turbulent incompressible fluid flow in a circular pipe with rough walls.     

A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilevered pipe in post-flutter region from macro to micro scale

A mathematical formulation is proposed to investigate the nonlinear flow-induced dynamic characteristics of a cantilevered pipe conveying fluid from macro to micro scale. The model is developed by using the extended Hamilton's principle in conjunction with the inextensibility condition and laminar and turbulent flow profiles as well as modified couple stress theory. The current model is capable of recovering the classical model of cantilevered pipe conveying fluid by neglecting the couple stress effect. The governing equation of motion is presented in dimensionless form in a convenient and usable manner. To solve the problem at hand, the integro-partial-differential equation of motion is discretized into a set of ordinary differential equations via Galerkin method. Afterward, a Runge–Kutta's finite difference scheme is employed to evaluate the nonlinear dynamic response of the cantilevered pipe conveying fluid. A parametric study is carried out to examine the influences of mass parameter and dimensionless mean flow velocity on the nonlinear dynamic characteristics of the cantilevered pipe conveying fluid in post-flutter region. The role of size-dependency in the nonlinear behavior of pipe is explored by converting the new set of dimensionless parameters into the conventional one. Eventually, some convergence studies are performed to indicate the reliability of present results.     

Vibration and enhanced stability properties of fluid-conveying pipes with two symmetric elbows fitted at downstream end

In this paper, the vibration and stability properties of fluid-conveying pipes with two symmetric elbows fitted at downstream end are investigated. The fluid, after entering from the upstream end, is pushed downwards and eventually exits from the downstream end fitted with two symmetric elbows. The equation of motion is solved by means of Galerkin’s method with a four-mode approximation. Calculations are conducted for cantilevered and also for pinned–pinned slender pipes. It is found that the stability of the pipe system can be greatly enhanced with such downstream elbows. The vibration frequency of the fluid-conveying pipes can be comfortably controlled due to the downstream elbows with a selection of angle of inclination. The proposed geometry configuration of fluid-conveying pipes may be useful for the design and improvement of engineering pipeline systems and fluidic devices.     

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.      

FLUTTER INSTABILITY OF SUPPORTED PIPES CONVEYING FLUID SUBJECTED TO DISTRIBUTED FOLLOWER FORCES

In the past decades,it has been reported that divergence is the expected form of instability for fluid-conveying pipes with both ends supported.In this paper,the form of instability of supported pipes ...     

Results of direct numerical simulation of turbulent flow in a pipe of elliptical cross-section

Within the framework of the complete Navier-Stokes equations the turbulent flow in a pipe of elliptical cross-section with semiaxis ratio b/a = 0.5 is directly calculated for the Reynolds number Re = 6000 determined from the mean-flow velocity and the hydraulic diameter. The distribution of the average and pulsatory flow characteristics over the pipe cross-section are obtained. In particular, the secondary flow in the cross-section plane, typical of turbulent flows in noncircular pipes, is calculated. The equation for the longitudinal vorticity which determines the shape and intensity of the secondary flow is analyzed. In the balance equation for the pulsation kinetic energy the behavior of all the terms that characterize energy production, dissipation and redistribution over the pipe cross-section is described.     

Measurement of laminar,transitional and turbulent pipe flow using Stereoscopic-PIV

Stereoscopic particle image velocimetry (SPIV) is applied to measure the instantaneous three component velocity field of pipe flow over the full circular cross-section of the pipe. The light sheet is oriented perpendicular to the main flow direction, and therefore the flow structures are advected through the measurement plane by the mean flow. Applying Taylor’s hypothesis, the 3D flow field is reconstructed from the sequence of recorded vector fields. The large out-of-plane motion in this configuration puts a strong constraint on the recorded particle displacements, which limits the measurement accuracy. The light sheet thickness becomes an important parameter that determines the balance between the spatial resolution and signal to noise ratio. It is further demonstrated that so-called registration errors, which result from a small misalignment between the laser light sheet and the calibration target, easily become the predominant error in SPIV measurements. Measurements in laminar and turbulent pipe flow are compared to well established direct numerical simulations, and the accuracy of the instantaneous velocity vectors is found to be better than 1% of the mean axial velocity. This is sufficient to resolve the secondary flow patterns in transitional pipe flow, which are an order of magnitude smaller than the mean flow.     

Pulsatory Turbulent Compressible Fluid Flow and Propagation of Pressure Waves in a Channel

The properties of the damping coefficient and phase velocity of propagation of small-amplitude pressure waves as functions of the oscillation frequency are investigated for the turbulent flow of a weakly compressible fluid in a circular pipe. The wall friction is found by solving numerically the equation of motion and the relaxation equations for the turbulent shear stress and viscosity which provide the basis for a turbulent transfer model developed for unsteady conditions. The properties are explained in terms of an analysis of the calculated data on turbulent transfer. The results obtained are compared with experiments.     

Calculation of the stability of the flow in a circular pipe with injection

An experimental investigation of the transition of a laminar flow regime into a turbulent one has been carried out in [1] for a flow in a circular pipe which is organized due to injection through the porous lateral surface with a jammed leading end of the pipe. It was established as a result that injection leads to an increase in stability of the laminar flow regime and increases the Reynolds number of the transition to 10,000 instead of the value 2300 which is characteristic of flow in a circular pipe with impenetrable walls. A similar effect was discovered in [2], in which it was also obtained that the Reynolds number of stability loss under the action of injection can take values significantly larger than in pipes with impenetrable walls. The phenomenon of relaminarization of a turbulent flow in the initial section of a circular pipe under the action of injection has been experimentally detected at the entrance for relatively low Reynolds numbers in [3, 4]. Theoretical investigations of stability of flow with injection have been performed only for a plane channel [5, 6]. A calculation is made in this paper of the stability of a hydrodynamically developed flow in a circular pipe with injection through a porous lateral surface.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 82–86, May–June, 1984.     

A numerical study of bingham turbulent flow in sudden-expansion straight circular pipe

I.IntroductionBinghamfluidisonebranchofnon-Newtonianfluid,suchascrudeparaffinoil,highsediment--ladenwaterflow,highconcentrationmudandthelikewhicharetransportedinpipelinesinmanyindustries,soit'sofgreatsignificancetostudytheflowmechanismsofBinghamfluid.Tsaietal.II]studiedthelinkagebetweenBinghamfluidandpluggedflow.Wangetal.I2]measuredtheturbulencestructureofBinghammud.Mengetal.[3]researchedthekineticenergycorrectionfactorofBinghamfluidinacircularpipe.However,thestudyofBinghamfluidsofarisn't…     

Nonlinear dynamics of cantilevered pipes conveying fluid: Towards a further understanding of the effect of loose constraints

The nonlinear dynamics of a fluid-conveying cantilevered pipe with loose constraints placed somewhere along its length is investigated. The main objective of this study is to determine the effects of several geometrical and physical parameters of the loose constraints on the characteristics and behavior of pipes conveying fluid. Based on the full nonlinear equation of motion, the dynamical behavior of the pipe system is investigated. Phase portraits and bifurcation diagrams are constructed for a selected set of system parameters. Typical results are firstly compared to numerical ones reported previously and excellent agreement is obtained. Then, the threshold flow velocities for several key bifurcations including pitchfork, period doubling, chaos, and sticking behaviors are predicted, showing that in many cases, the gap size, stiffness, and asymmetry of the loose constraints have remarkable effects on the nonlinear responses of the cantilevered pipe conveying fluid. For a pipe system with small/large constraint gap sizes, small constraint stiffness, or large constraint offset, some of the complex dynamical behaviors including chaos and period-doubling bifurcations would disappear, at least in the flow velocity range of interest.     

Stability analysis of viscoelastic curved pipes conveying fluid

Based on the Hamilton' s principle for elastic systems of changing mass, a differential equation of motion for viscoelastic curved pipes conveying fluid was derived using variational method, and the complex characteristic equation for the viscoelastic circular pipe conveying fluid was obtained by normalized power series method. The effects of dimensionless delay time on the variation relationship between dimensionless complex frequency of the clamped-clamped viscoelastic circular pipe conveying fluid with the Kelvin-Voigt model and dimensionless flow velocity were analyzed. For greater dimensionless delay time, the behavior of the viscoelastic pipe is that the first, second and third mode does not couple, while the pipe behaves divergent instability in the first and second order mode, then single-mode flutter takes place in the first order mode.     

Effect of the pattern of initial perturbations on the steady pipe flow regime

The influence of the inlet flow formation mode on the steady flow regime in a circular pipe has been investigated experimentally. For a given inlet flow formation mode the Reynolds number Re* at which the transition from laminar to turbulent steady flow occurred was determined. With decrease in the Reynolds number the difference between the resistance coefficients for laminar and turbulent flows decreases. At a Reynolds number approximately equal to 1000 the resistance coefficients calculated from the Hagen-Poiseuille formula for laminar steady flow and from the Prandtl formula for turbulent steady flow are equal. Therefore, we may assume that at Re > 1000 steady pipe flow can only be laminar and in this case it is meaningless to speak of a transition from one steady pipe flow regime to the other. The previously published results [1–9] show that the Reynolds number at which laminar goes over into turbulent steady flow decreases with increase in the intensity of the inlet pulsations. However, at the highest inlet pulsation intensities realized experimentally, turbulent flow was observed only at Reynolds numbers higher than a certain value, which in different experiments varied over the range 1900–2320 [10]. In spite of this scatter, it has been assumed that in the experiments a so-called lower critical Reynolds number was determined, such that at higher Reynolds numbers turbulent flow can be observed and at lower Reynolds numbers for any inlet perturbations only steady laminar flow can be realized. In contrast to the lower critical Reynolds number, the Re* values obtained in the present study, were determined for given (not arbitrary) inlet flow formation modes. In this study, it is experimentally shown that the Re* values depend not only on the pipe inlet pulsation intensity but also on the pulsation flow pattern. This result suggests that in the previous experiments the Re* values were determined and that their scatter is related with the different pulsation flow patterns at the pipe inlet. The experimental data so far obtained are insufficient either to determine the lower critical Reynolds number or even to assert that this number exists for a pipe at all.     

A Constitutive Explanation of Manning’s Formula

Manning’s empirical formula for evaluating the mean velocity for a steady uniform turbulent flow in pressure conduits with circular cross-section and in a wide rectangular open channel, can be theoretically justified by introducing a virtual viscosity or mixing turbulent coefficient, depending on the velocity and the position of a particle according to Boussinesq’s hypothesis for the turbulent flow in a pipe. The balance equation yields an ordinary differential equation whose integration yield the distribution of the velocity.