Analytical predictions for vibration phase shifts along fluid-conveying pipes due to Coriolis forces and imperfections

Resonant vibrations of a fluid-conveying pipe are investigated, with special consideration to axial shifts in vibration phase accompanying fluid flow and various imperfections. This is relevant for understanding elastic wave propagation in general, and for the design and trouble-shooting of phase-shift measuring devices such as Coriolis mass flowmeters in particular. Small imperfections related to elastic and dissipative support conditions are specifically addressed, but the suggested approach is readily applicable to other kinds of imperfection, e.g. non-uniform stiffness or mass, non-proportional damping, weak nonlinearity, and flow pulsation. A multiple time scaling perturbation analysis is employed for a simple model of an imperfect fluid-conveying pipe. This leads to simple analytical expressions for the approximate prediction of phase shift, providing direct insight into which imperfections affect phase shift, and in which manner. The analytical predictions are tested against results obtained by pure numerical analysis using a Galerkin expansion, showing very good agreement. For small imperfections the analytical predictions are thus comparable in accuracy to numerical simulation, but provide much more insight. This may aid in creating practically useful hypotheses that hold more generally for real systems of complex geometry, e.g. that asymmetry or non-proportionality in axial distribution of damping will induce phase shifts in a manner similar to that of fluid flow, while the symmetric part of damping as well as non-uniformity in mass or stiffness do not affect phase shift. The validity of such hypotheses can be tested using detailed fluid-structure interaction computer models or laboratory experiments.     

Global dynamics of a pipe conveying pulsating fluid in primary parametrical resonance: Analytical and numerical results from the nonlinear wave equation

The chaotic dynamics of nonlinear waves in the harmonic-forced fluid-conveying pipe in primary parametrical resonance, is explored analytically and numerically. The multiple scale method is applied to obtain an equivalent nonlinear wave equation from the complicated nonlinear governing equation describing the fluid conveyed in a pipe. With the Melnikov method, the persistence of a heteroclinic structure is shown to be satisfied and its condition is given in functional form. Similarly, for the heteroclinic orbit, using geometric analysis, a condition function of the stable manifold is derived for the orbit to return to the stable manifold from the saddle point. The persistent homoclinic structures and threshold of chaos in the Smale-horseshoe sense are obtained for the fluid-conveying pipe under both conditions, indicating how the external excitation amplitude can change substantially the global dynamics of the fluid conveyed in the pipe. A numerical approach was used to test the prediction from theory. The impact of the external excitation amplitude on the nonlinear wave in the fluid-conveying pipe was also studied from numerical simulations. Both theoretical predications and numerical simulations attest to the complex chaotic motion of fluid-conveying pipes.     

Analysis of pipeline networks using two-ports

Majority of vibration problems arise in pipeline networks are attributed to the high-pressure pulsations. Pulsations are generated by fluid machines such as compressors and pumps. These pulsations turn into shaking forces at elements such as pipe bends and pipe reducers, which in turn excite vibrations in the connected piping network. High vibrations beyond the endurance limit of the pipe material may cause damage to pipes, supports, and equipment. In addition, if the source pulsation frequency coincides with one of the natural frequencies of the piping network, resonance will take place and the vibrations will be magnified to a large scale. Obviously, if these vibrations are not well controlled, they might cause damage to the whole system and foundation, and might lead to substantial financial losses. Thus, prediction of pulsations is important for safe and proper operation. In this paper, a pilot plant equipped with a reciprocating compressor, pipes, bends, and terminated by a vessel is built. The network is modeled using the two-port theory that splits the network into several cascaded elements, and predicts the response of the network. The prediction model uses the measured compressor source data as an input, which is determined by the indirect multi-load method that is usually used to characterize internal combustion engines. A pulsation suppression device is designed, modeled, manufactured and inserted into the pilot network. The pressure pulsations are measured with and without the pulsation suppression device, and compared to the predictions using the two-port theory.     

脉动流在分叉管中通栓效果的晶格玻尔兹曼方法研究

血栓引发的各种心血管疾病一直威胁着人们的健康. 在已经产生血栓的血管中, 脉动对于疏通血管有良好的作用. 由于血液的黏滞作用以及红细胞的惯性, 脉动流的频率会影响血管通栓的效果. 在分叉管模型中, 低压差的条件下, 由于另一畅通管子的导通作用减少了回流, 导致通栓效果不理想. 通过增大压差和提高脉动流的振幅, 降低畅通管子导通作用的影响, 研究脉动流在分叉管中的通栓效果. 研究发现, 脉动低频通栓效果好, 但是通栓需要的时间较长; 高频通栓时间短, 但是当频率高于一定值, 则通栓效果不明显. 细胞和管壁的摩察系数对通栓效果也有影响.     

Heat transfer analysis of thermal energy storage using phase change materials

An analytical study is described of a freezing/melting phenomenon of a phase change material encapsulated in a pipe with the heat transfer fluid flowing in the annulus of a concentric pipe. The moving front problem is solved numerically using the boundary immobilization technique. Parameters such as the Stefan number, the Fourier number and the diameter ratio of the two concentric pipes to obtain sufficiently large heat transfer rates to make the system suitable for thermal energy storage are examined.     

Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass

The three-dimensional (3-D) dynamics of a fluid-conveying cantilevered pipe fitted with an end-mass and additional intra-span spring-support is investigated in this paper, both theoretically and experimentally. The main objective is to examine how the dynamics of a cantilevered pipe with additional spring-support is modified by the presence of a small mass attached at the free end. In the theoretical study, the nonlinear three-dimensional equations of motion are discretized via Galerkin's method, and the resulting equations are solved by a finite difference method. For the cases studied, the system was found to lose stability by planar flutter; as the flow velocity is increased beyond that point, a sequence of higher order bifurcations ensue, involving 2-D and 3-D periodic and quasiperiodic motions, as well as chaotic ones. In the experiments, performed with elastomer pipes and water flow, similarly 2-D and 3-D periodic, quasiperiodic and chaotic oscillations were observed. Theory and experiments have been shown to be in good qualitative and quantitative agreement. The experimental behaviour is illustrated by video clips (electronic annexes). Moreover, the effects of (i) small stiffness imperfections and (ii) excitation by a point-force are explored theoretically in a preliminary way.     

A study of vibroacoustic coupling between a pump and attached water-filled pipes

This paper presents a model for the vibroacoustical behavior of a pump coupled with water-filled pipes. Coupling between (a) the pump and the inlet and outlet pipes, and (b) the pipe wall and the fluid contained in the pipe, is investigated through analytical modeling and numerical simulation. In the model, the pump is represented by a rigid body supported by multiple elastic mounts, and the inlet and outlet pipes by two semi-infinite water-filled pipes. The vibration characteristics of the coupled system under the excitation of mechanical forces and fluid-borne forces at the pump are calculated. The results enhance our understanding about how the input mechanical and fluid excitation energy at the pump is transmitted to the pipes and how to relate the piping vibroacoustical response to the excitations at the pump. This study assists in predicting dynamic stress in pipes for given excitations at the pump, and in developing methods to identify the nature (fluid or mechanical) of the excitation forces at the pump using the vibration and dynamic pressure measurements on the pump/pipe system.     

Investigation on Rijke pipe＇s acoustic characteristics by numerical simulation： modeling the pulsing flow field coupled the inner of pipe with its outer space

Based on the results of fluid dynamics, heat transfer and acoustics, a Computational Fluid Dynamics （CFD） method was utilized to study the acoustic characteristics and self-excited pulsation mechanism inside a Rijke pipe. To avoid settling the irrational boundary conditions of the finite-amplitude standing wave in the Rijke thermo-acoustic system, the simulation modeling in the flow field, which coupled the inner of pipe with its outer space, was carried out to replace the traditional way in form of internal flow field numerical investigations. A hypothesis for heat source in energy equation including the relationship on unsteady heat of air around heat source, oscillation pressure and oscillation velocity was presented. To reflect the essence of Rijke pipe, simulation on self-excited oscillation was conducted by means of its own pulsation of pressure, velocity and temperature. This method can make the convergence process steady and effectively avoid divergence. The physical phenomenon of the self-excited Rijke pipe was analyzed. Moreover, the mechanisms on the Rijke pipe＇s self-excited oscillation were explained. Based on this method, comparative researches on the acoustic characteristic of the Rijke pipe with different size and different shape of nozzle were performed. The simulation results agreed with the experimental data satisfactorily. The results show that this numerical simulation can be used to study the sound pressure of nozzle for the engineering application of Rijke pipes.     

Postbuckling and vibration of end-supported elastica pipes conveying fluid and columns under follower loads

Fluid-conveying pipes with supported ends buckle when the fluid velocity reaches a critical value. For higher velocities, the postbuckled equilibrium shape can be directly related to that for a column under a follower end load. However, the corresponding vibration frequencies are different due to the Coriolis force associated with the fluid flow. Clamped–clamped, pinned–pinned, and clamped–pinned pipes are considered first. Axial sliding is permitted at the downstream end. The pipe is modeled as an inextensible elastica. The equilibrium shape may have large displacements, and small motions about that shape are analyzed. The behavior is conservative in the prebuckling range and nonconservative in the postbuckling range (during which the Coriolis force does work and the motions decay). Next, related columns are studied, first with a concentrated follower load at the axially sliding end, and then with a distributed follower load. In all cases, a shooting method is used to solve the nonlinear boundary-value problem for the equilibrium configuration, and to solve the linear boundary-value problem for the first four vibration frequencies. The results for the three different types of loading are compared.     

On the effects of reflections on time delay estimation for leak detection in buried plastic water pipes

This paper is concerned with the way in which wave reflections in a fluid-filled pipe affect the cross-correlation function of two leak noise signals used to detect and locate leaks in buried water pipes. Propagating waves generated by leak noise reverberate in a pipe network system, as they encounter features such as changes in section, and resistance such as valves, and pipe junctions. A theoretical model of a straight pipe with discontinuities, which cause reflections, is developed and incorporated into a model of the cross-correlation function. The reasons why the reflections and the low-pass filtering properties of the pipe can be largely removed by the generalised cross-correlation (GCC) phase transform (PHAT) are determined. Using the analytical model, theoretical predictions of the basic cross-correlation function (BCC) and the GCC PHAT are compared with experimental data from a specially constructed test site in Canada.     

螺旋型波纹管流阻测量系统研制及液氮压降测量

超导电缆芯通常内嵌于真空绝热波纹管并被管内低温流动介质冷却和保护。螺旋型波纹管因一次成型制作长度上的优势更适合于大长度超导电缆应用。设计并搭建了螺旋型波纹管液氮流动压降特性实验台,不仅可以方便地更换被测波纹管样品,而且允许插入不同规格的线芯模拟物。利用该实验台测量了液氮流量1～9 L/min区间内不同规格(通径11～15 mm)螺旋型波纹管插入4 mm线芯后的流动压降特性。实验结果验证了三维波纹管压降损失数值模型的准确性。同时,通过进行不同尺寸波纹管实验,发现尺寸变化对摩擦因子变化规律的影响不明显,这为通过该实验台获得小管径实验数据用于指导大管径实际应用波纹管设计提供了理论基础。     

Robustness of Objective and Subjective Speckle Patterns in the Speckle Strain Gauge

The robustness to rigid body object motions of three optical systems used in the speckle strain gauge were experimentally investigated and compared with analytical results of the correlation. It was found that an out-of-plane motion of the object damaged the reliability of the strain measure when recording the objective speckle patterns while subjective speckle patterns were more robust. Besides out-of-plane object motions, the robustness of a free-space geometry and an afocal imaging configuration are approximately the same, while a telecentric imaging system is more robust to rigid body motions but more sensitive to deformation gradients (basically in-plane rotation and tilt). Results from a measurement of the relaxation in a lead-tin alloy used in organ pipes is also presented.Presented at 1996 International Workshop on Interferometry (IWI ‘96), August 27-29, Wako, Saitama, Japan.     

New Equation for Predicting Pipe Friction Coefficients Using the Statistical Based Entropy Concepts

In general, this new equation is significant for designing and operating a pipeline to predict flow discharge. In order to predict the flow discharge, accurate determination of the flow loss due to pipe friction is very important. However, existing pipe friction coefficient equations have difficulties in obtaining key variables or those only applicable to pipes with specific conditions. Thus, this study develops a new equation for predicting pipe friction coefficients using statistically based entropy concepts, which are currently being used in various fields. The parameters in the proposed equation can be easily obtained and are easy to estimate. Existing formulas for calculating pipe friction coefficient requires the friction head loss and Reynolds number. Unlike existing formulas, the proposed equation only requires pipe specifications, entropy value and average velocity. The developed equation can predict the friction coefficient by using the well-known entropy, the mean velocity and the pipe specifications. The comparison results with the Nikuradse’s experimental data show that the R2 and RMSE values were 0.998 and 0.000366 in smooth pipe, and 0.979 to 0.994 or 0.000399 to 0.000436 in rough pipe, and the discrepancy ratio analysis results show that the accuracy of both results in smooth and rough pipes is very close to zero. The proposed equation will enable the easier estimation of flow rates.     

Formulas for compression wave and oscillating flow in circular pipe

This paper shows analytical solutions for compression wave and for steady-state oscillating flow in a pipe with a circular cross section. We considered the compression wave to be one of fluid phenomena, and paid our attention to the equation of fluid motion and of continuity. Pressure and flow velocity are directly treated in analytical solutions. Not only either of them but also flow rate may be chosen as an input and outputs by easy rearrangement of the formulas. This paper enables us to express various behavior of the compression wave by using analytical solutions with the phase velocity and the spatial absorption coefficient. Those analytical solutions enable us to reduce computation time and analysis cost on not only duct acoustics but also water hammer and combustion instabilities. Of course, our formulas may contribute to designing of microdevices such as MEMS strongly influenced by wall-viscosity.     

On the scattering of torsional elastic waves from axisymmetric defects in coated pipes

Long range ultrasonic testing is now a well established method for examining in-service degradation in pipelines. In order to protect pipelines from the surrounding environment it is common for viscoelastic coatings to be applied to the outer surface. These coatings are, however, known to impact on the ability of long range ultrasonic techniques to locate degradation, or defects, within a coated pipe. The coating dissipates sound energy travelling along the pipe, attenuating both the incident and reflected signals making responses from defects difficult to detect. This article aims to investigate the influence of a viscoelastic coating on the ability of long range ultrasonic testing to detect a defect in an axisymmetric pipe. The article focuses on understanding the behaviour of the fundamental torsional mode and quantifying the effect of bitumen coatings on reflection coefficients generated by axisymmetric defects. Reflection coefficients are measured experimentally for coated and uncoated pipes and compared to theoretical predictions generated using numerical mode matching and a hybrid finite element technique. Good agreement between prediction and measurement is observed for uncoated pipes, and it is shown that the theoretical methods presented here are fast and efficient making them suitable for studying long pipe runs. However, when studying coated pipes agreement between theory and prediction is observed to be poor for predictions based on those bulk acoustic properties currently reported in the literature for bitumen. Good agreement is observed only after conducting a parametric study to identify more appropriate values for the bulk acoustic properties. Furthermore, the reflection coefficients obtained for the fundamental torsional mode in a coated pipe show that significant sound attenuation is present over relatively short lengths of coating, thus quantifying those problems commonly encountered with the use of long range ultrasonic testing on coated pipes in the field.     

Roughness of organ pipe sound due to frequency comb

In the sound spectrum of flue organ pipes in addition to the usual harmonic partials, sometimes a series of equidistant but not harmonic lines can be found. This phenomenon has been observed in the recorded sound of pipes from different pipe ranks. The second set of spectral lines is similar to "frequency combs" used in optics for accurate measurement of optical frequencies. Analysis of measured sound spectra with and without frequency comb and simulations are presented and discussed in the paper. The appearance of frequency combs in the sound spectrum is explained by a model that assumes the presence of a mouth tone in addition to the pipe sound. Mouth tone bursts are generated when the oscillating air jet passes the upper lip. The burst repetition frequency is locked to the fundamental frequency of the pipe and the bursts are coherent with a pulse-to-pulse phase shift. The phase shift explains the observed frequency offset of the frequency comb to the harmonic frequencies. The simulations also show that weak and fluctuating mouth tones cannot generate frequency comb due to a lack of coherence.     

Elastic fields of a screw superdislocation with a hollow core (pipe) perpendicular to the free crystal surface

Screw superdislocations with hollow cores (pipes) which arise during growth of semiconductor crystals such as silicon carbide and gallium nitride are considered. Exact analytical expressions are first derived for the displacements, strains, and stresses associated with a pipe oriented perpendicular to the free planar surface of an elastically isotropic half-space. It is shown that the stress field of the dislocation present in the pipe is heavily affected when the exact boundary conditions at the free cylindrical surface of the pipe are taken into account. The influence is the strongest in the region around the pipe at distances of the order of the pipe radius from the surface of the pipe. In this region, the elastic strains can be as large as a few tenths of one percent. The results obtained can be useful in analyzing the interaction of pipes with one another and with other defects, as well as in simulating the behavior of pipes during crystal growth.     

Sound transmission in pipes with porous walls

Pipes with porous (permeable) walls have received the attention of several authors as a noise control element in automotive intake systems; however, a closed theory of sound transmission including the effect of the coupling of the internal and external acoustic fields and the presence of mean flow does not appear to be available. The present paper proposes an integro-differential system for the propagation of plane sound waves in pipes with porous walls, and presents its general numerical solution, as well as an approximate analytical solution. The predicted effect of the coupling between the internal and external acoustic fields in a circular pipe made of reinforced woven fabric walls is shown, and the transmission loss predictions are compared with the existing experimental data in the literature.     

Application of the transfer matrix method to non-conservative systems involving fluid flow in curved pipes

The application of the transfer matrix method to problems of fluid flow in planar curved pipes is presented. Also a presentation is given of how to deal with the stability of non-conservative problems when using the dynamic method. In the method proposed, account is taken of the effect of the initial state of stress on the dynamic behaviour of the pipe. Three well-known problems are solved to verify the convergence and applicability of the transfer matrix formulation to non-conservative problems involving fluid flow.