Axisymmetric fluid-dominated wave in fluid-filled plastic pipes: Loading effects of surrounding elastic medium

Axisymmetric (n = 0) waves that propagate at low frequencies are of practical interest in the application of acoustic techniques for the detection of leaks in fluid-filled pipelines. A general expression for the fluid-dominated (s = 1) wavenumber is presented in a thin-walled fluid-filled pipe surrounded by an elastic medium. In this paper the analysis is extended to investigate the loading effects of surrounding medium on the low-frequency propagation characteristics of the s = 1 wave. The analytical model is subsequently applied to MDPE water pipes surrounding by three media, namely an air, water and soil. It is used to demonstrate explicitly the loading effects of surrounding medium, acting as a combination of mass, stiffness and radiation damping on the s = 1 wavenumber. Good agreement is achieved between the measurements and predictions. The theory with experimental validations provides the basis for improving acoustic leak detection methods in fluid-filled pipe systems.     

Modelling of linear wave propagation in spatial fluid filled pipe systems consisting of elastic curved and straight elements

This paper is concerned with the theoretical analysis of time harmonic dynamics of compound elastic pipes with and without internal fluid loading. Compound pipes are assembled as a sequence of segments, each of which has a constant curvature. As a prerequisite for the wave propagation analysis, dispersion equations are solved, Green’s matrices are formulated and Somigliana’s identities are derived for an isolated curved segment. The governing equations of wave motion of a compound pipe are obtained as an ensemble of the boundary integral equations for individual segments and the continuity conditions at their interfaces. The proposed methodology is validated in several benchmark problems and then applied for analysis of the periodicity effects. The results obtained for piping systems with a variable number of identical curved segments are put into the context of the classical Floquet theory. Brief parametric studies suggest that the curved inserts can be employed as a tool for the passive control of wave propagation in fluid-filled pipes, and their stop band characteristics may be tailored to reach desirable attenuation levels in prescribed frequency ranges.     

黏弹介质包裹的充液管道中低频轴对称波传播特性

研究埋地充液管道中低频轴对称波传播特性。将土壤考虑为黏弹介质,结合Kennard薄壳方程和Kelvin-Voigt线性黏弹性模型,引入土壤载荷矩阵,推导出土-管滑移情形下流体主导波和管壁压缩波的相速度表达式。通过数值模拟计算得到流体主导波和管壁压缩波的频散和衰减曲线并进行可靠性验证,分析两种波引起的管壁径向位移之比,讨论厚径比和品质因子对流体主导波传播的影响。结果表明,黏弹介质对流体主导波和管壁压缩波的相速度影响较小,但对衰减影响较大;流体主导波对管壁径向位移有较大的影响,是泄露噪声传播的主要载体;厚径比越大,流体主导波的相速度越大,衰减越小;而品质因子越大,流体主导波的频散和衰减都越小。研究结果可为埋地充液管道的泄漏检测提供一定的理论参考。      

Interaction of low-frequency axisymmetric ultrasonic guided waves with bends in pipes of arbitrary bend angle and general bend radius

The use of ultrasonic guided waves for the inspection of pipes with elbow and U-type bends has received much attention in recent years, but studies for more general bend angles which may also occur commonly, for example in cross-country pipes, are limited. Here, we address this topic considering a general bend angle φ, a more general mean bend radius R in terms of the wavelength of the mode studied and pipe thickness b. We use 3D Finite Element (FE) simulation to understand the propagation of fundamental axisymmetric L(0, 2) mode across bends of different angles φ. The effect of the ratio of the mean bend radius to the wavelength of the mode studied, on the transmission and reflection of incident wave is also considered. The studies show that as the bend angle is reduced, a progressively larger extent of mode-conversion affects the transmission and velocity characteristics of the L(0, 2) mode. However the overall message on the potential of guided waves for inspection and monitoring of bent pipes remains positive, as bends seem to impact mode transmission only to the extent of 20% even at low bend angles. The conclusions seem to be valid for different typical pipe thicknesses b and bend radii. The modeling approach is validated by experiments and discussed in light of physics of guided waves.     

Vibration characteristic analysis of buried pipes using the wave propagation approach

A theoretical analysis for the free vibration of simply supported buried pipes has been investigated using the wave propagation approach. The pipe modeled as a thin cylindrical shell of linear homogeneous isotropic elastic material buried in a linear isotropic homogeneous elastic medium of infinite extent. The vibrations of the pipe are examined by using Flüggle shell equation. The natural frequencies are obtained for the pipes surrounded by vacuo or elastic medium. The results are compared with those available in the literature and agreement is found with them. It is found that the free vibration frequency of the pipe does not appear for some of the axial or circular vibration modes and the real natural frequencies of the pipe are significantly dependent on the rigidity of the surrounding medium.     

Guided waves dispersion equations for orthotropic multilayered pipes solved using standard finite elements code

The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0, n), torsional T(0, n) and flexural F(m, n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method.     

Flow visualization of relaminarization phenomena in curved pipes and the related measurements

Flow visualization results for secondary flow phenomena at the exit of 90° and 180° bends and helically coiled pipes (1, 2 and 5 turns), (radius of curvatureR _c=381 mm, inside pipe diameterd=37.5 mm, curvature ratiod/2R _c=0.049) and in the downstream straight pipe (l/d=30) are presented to study the stabilizing (relaminarization) effect in curved pipes with fully developed entry turbulent air flow and the destabilizing (re-transition from laminar to turbulent flow) effect in the downstream straight region. The entry Reynolds numbers areRe=2200, 3200, 4300 and 5300). The related measurement results using a hot-film anemometer are presented for developing profiles of the time-mean streamwise velocity distribution and the axial turbulence intensity field in the 180° return bend and in the downstream straight pipe for Reynolds numbersRe=3200, 4300, 6300 and 8200. The time traces showing the output of the hot-film sensor are also presented for developing fluctuating velocity field in the 180° bend and in the downstream straight pipe for the same Reynolds number range. Some aspects of the relaminarization phenomena in curved pipes and the re-transition phenomena from laminar to turbulent flow in the downstream straight pipe are clarified by the present experimental investigation.     

EXPERIMENTAL STUDY ON THE VORTEX-INDUCED VIBRATION OF TOWED PIPES

We experimentally attempted to understand the vibration characteristics of a flexible pipe excited by vortex shedding. This has been extensively studied in the previous decades (for example, see Sarpkaya 1979 Journal of Applied Mechanics46, 241-258; Price et al. 1989 Eighth International Conference on Offshore Mechanics and Arctic Engineering, The Hague-March 19 -23, 447-454; Yoerger et al. 1991 Journal of Offshore Mechanics and Arctic Engineering, Transaction of Engineers113, 117-127; Grosenbaugh et al. 1991Journal of Offshore Mechanics and Arctic Engineering, Transaction of Engineers113 , 199-204; Brika and Laneville 1992 Journal of Fluid Mechanics250, 481-508; Chakrabarti et al. 1993 Ocean Engineering20, 135-162; Jong 1983 Ph.D. Dissertation, Department of Ocean Engineering, M. I. T.; Kimet al. 1986 Journal of Energy Resources Technology, Transactions of American Society of Mechanical Engineers108, 77-83). However, there are still areas that need more study. One of them is the relation between spatial characteristics of a flow-induced vibrating pipe, such as its length, the distribution of wave number, and frequency responses. A non-linear mechanism between the responses of in-line and cross-flow directions is also an area of interest, if the pipe is relatively long so that structural modal density is reasonably high. In order to investigate such areas, two kinds of instrumented pipe were designed. The instrumented pipes, of which the lengths are equally 6 m, are wound with rubber and silicon tape in different ways, having different vortex-shedding conditions. One has uniform cross-section of diameter of 26·7 mm, and the other has equally spaced four sub-sections, which are composed of different diameters of 75·9, 61·1, 45·6 and 26·7 mm. Both pipes are towed in a water tank (200 m×16 m×7 m) so that they experienced different vortex-shedding excitations. Various measures were obtained from the towing experiment, including frequency responses, the time-domain tracing of in-line and cross-flow responses, and Wigner-Ville distributions. The experimental results analyzed by using these measures exhibit several valuable features. One of them is that the natural frequencies and their corresponding strain mode shapes dominate the strain response of the uniform pipe. However for those of non-uniform pipe, the responses are more likely local and many modes participate in it.     

Nonlinear and linear wave phenomena in narrow pipes

Phenomena arising in the course of wave propagation in narrow pipes are considered. For nonlinear waves described by the generalized Webster equation, a simplified nonlinear equation is obtained that allows for low-frequency geometric dispersion causing an asymmetric distortion of the periodic wave profile, which qualitatively resembles the distortion of a nonlinear wave in a diffracted beam. Tunneling of a wave through a pipe constriction is investigated. Possible applications of the phenomenon are discussed, and its relation to the problems of quantum mechanics because of the similarity of the basic equations of the Klein-Gordon and Schrödinger types is pointed out. The importance of studying the tunneling of nonlinear waves and broadband signals is indicated.     

Propagation characteristics of guided waves in a liquid filled pipe embedded in a porous medium

A model of a liquid-filled pipe embedded in a porous medium is built to research its wave propagation characteristics,and to analyze the effect of the parameters of porous media on the dispersion.The dispersion equations are established on the basis of the elastic-dynamic theory of the liquid-saturated porous solid.The characteristic of dispersion and the time domain waveform in pipes of different thicknesses and with different porous-medium parameters are discussed theoretically and numerically.Results reveal that the porosity has little impact on dispersion,and the attenuation of guided wave increases with porosity,whilst the porosity influences the displacement amplitude of the time domain waveform.It is hard to detect the permeability variation of the media by the dispersion.The drawn conclusion can provide some theoretical instruction and guidance for the nondestructive testing of buried pipe.     

An actuator for the n=2 circumferential mode of a pipe

The axial propagating wave of the n=2 circumferential mode in a pipe can cause large vibration when the wave cuts on and may cause fatigue damage or transmit structure-borne noise along the pipe. For this reason, a control system (passive or active) is desirable. In this paper, a PZT modal actuator for the n=2 mode, which could be used in an active control system, is described. It is constructed from a set of PZT elements bonded to the pipe. By arranging them in form of the n=2 mode, only the motion of that particular mode is generated and is proportional to the applied voltage. With two PZT modal actuators, which are in forms of sine and cosine functions, the orientation of the propagating wave of the n=2 mode can be modified to any angle. In this paper a theoretical model for the actuator is developed and is validated by some experimental work.     

Shortened equations for the elliptically polarized wave

The field of an elliptically polarized wave is represented for the convenient solution of practical problems. The wave amplitude is written in terms of four parameters: magnitude E, phase φ, angle ψ of the polarization plane, and ellipticity ? (|?| is the ratio of the ellipse axes). The shortened equations are derived using these parameters and the solution of the shortened equations is obtained for a linear medium.     

管道弯头对低频纵向导波传播特性影响分析

管道弯头显著改变了导波传播特性,影响了对检测信号的解读,研究弯头对导波传播特性的影响是实现复杂管道系统导波检测的基础。采用半解析有限元法计算弯管导波频散曲线,分析了弯管导波频散曲线所呈现的不同特征,并基于弯管导波频散曲线,以低频L(0,1)模态导波为研究对象,实验研究了低频L(0,1)模态导波通过管道弯头时的模态变换特征。研究结果发现,当L(0,1)模态导波通过管道弯头时,不仅会发生L(0,1)到F(1,1)的模态变换,还会模态变换出反向L(0,1)模态导波,即弯头反射现象,且随着激励频率的降低和弯头弯曲半径的减小,弯头反射现象愈发明显。研究结果将深化对弯管导波传播特性的认识,推动导波检测技术在复杂管道系统检测中的应用。      

Simulation of the vector wave field of a low-frequency sound source in the ocean: Some important features

An approach to the simulation of low frequency vector wave fields in stratified media (mainly in the ocean) is considered. The approach is characterized by an improved stability with respect to dividing the medium into many layers of arbitrary thickness. The model for the sound field of a point source is based on an integral representation of two-dimensional, cylindrically symmetric vector wave fields in inhomogeneous media, so that the contributions of all types of waves are included automatically. The model medium is subdivided into N horizontally homogeneous layers for which 4(N?1) equations are formulated to satisfy the boundary conditions between adjacent layers. The method of the generalized Schmidt matrix is used to obtain the coefficients of the equations; these coefficients are substituted into the expressions (of the Fourier-Bessel integral type) for the local parameters of the field. The latter are calculated according to the numerical procedure, and the results are used to model the distributions of the acoustic pressure and the horizontal and vertical components of the particle velocity in liquid and elastic media. The instability of the calculation procedure may result in a disagreement between the model and the exact solution. However, the disagreement is shown to occur mainly in models containing excessively thick layers. A way for improving the stability of the numerical model is suggested. The simulation results are compared with the exact analytical solution for the simplest example and with the results obtained according to the commonly used generalized matrix procedure (the benchmark problem). The examples of the practical application of the model for investigating more complex seismoacoustic wave fields in the ocean are presented.     

Kinetic Alfvén wave turbulence in space plasmas

This work presents the derivation of nonlinear coupled equations for the evolution of solar wind turbulence. These equations are governing the coupled dynamics of kinetic Alfvén wave and ion acoustic wave. Numerical simulation of these equations is also presented. The ponderomotive nonlinearity is incorporated in the wave dynamics. Filamentation of kinetic Alfvén wave and the turbulent spectra are presented in intermediate-β plasmas at heliocentric distances (0.3 AU?r<1.0 AU). The growing filaments and steeper turbulent spectra (of power law k−S, 5/3?S?3) can be responsible for plasma heating and particle acceleration in solar wind.     

The effect of pipe bend on low-frequency longitudinal mode guided wave propagation

The pipe bend significantly changes the propagation characteristics of guided wave,and makes the interpretation of the received signals difficult.Therefore,better understanding of guided wave propagating in bended pipe is essential for the inspection of pipeline comprising bends.First of all,the different features of dispersion curves derived with the semi-analytical finite element method for guided wave in bended pipes are summarized.Secondly,based on the dispersion curves for guided wave in bended pipes,experiments are performed to investigate the mode conversions of L(0,1) mode guided wave traveling through pipe bends.It is found that,except for the mode conversion from L(0,1) to F(1,1),the L(0,1) reflections of bends are also observed in some cases,which are proven to be the mode converted negative L(0,1)mode guided wave,and the negative L(0,1) mode guided wave becomes more obvious with the decrease of excitation frequency and bending radius.The findings of this paper will provide some insight for guided wave behavior in bended pipe,and generalize the application of guided wave inspection in practical pipelines.     

Scattering of fermions on the SU(5) colorless magnetic monopole

We discuss the scattering of fermions (d?, u-quarks and e^c) on the SU(5) colorless magnetic monopole. The radial functions of scattering wave are totally separated from the angular parts in the Dirac equations which then reduce to a set of coupled ordinary differential equations. We solve these equations numerically for the d?e^c group with j = 1, and get the result that only the incoming wave of d with j = 1 and j′ = 0 can be scattered into e^c. The cross section is obtained which is intimately connected with the catalysis effect of the monopole for baryon decay.     

Application of the (G^{\prime}/G)-expansion method for the Burgers, Burgers–Huxley and modified Burgers–KdV equations

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.     

Optical parametric amplification beyond the slowly varying amplitude approximation

The coupled-wave equations describing optical parametric amplification (OPA) are usually solved in the slowly varying amplitude (SVA) approximation regime, in which the second-order derivatives of the signal and idler amplitudes are ignored and in fact the electromagnetic effects due to exit face of the medium is not involved. Here, an analytical plane-wave solution of these coupled-wave equations in a non-absorbing medium is presented. The solutions are derived beyond the SVA approximation up to order of κ/k (coupling constant over the wave number). The intensity distributions of the signal and the idler waves show a periodic behavior about their corresponding distributions of SVA-adapted solution. This behavior can be explained by the interference of the forward propagating signal (idler) wave and the corresponding backward one resulted from the reflection by the end face of the medium. Furthermore, this interference pattern in the medium can in turn serve as a periodic source for the next generations of the signal and idler waves. Therefore, the superposition of the waves, generated from different points of this periodic source, at the exit face of the medium shows an oscillatory behavior of the transmitted signal (idler) wave in terms of normalized coupling constant, κL. This study also shows that this effect is more considerable for high intensity pump beam, high relative refractive index and short length of the nonlinear medium.