Bifurcations to divergence and flutter in flow-induced oscillations: A finite dimensional analysis

The behaviours of a pipe conveying fluid and a fluid loaded panel are studied from the viewpoint of differentiable dynamics. Non-linear terms are included and it is shown how the partial differential equation of motion can be recast, by Galerkin's method and modal truncation, in the form of an ordinary differential equation in Euclidean n-space. This evolution equation is then analysed qualitatively, attention being paid to bifurcations which occur as the control parameters of axial force and flow velocity are varied. Bifurcations of fixed points occur when at least one of the eigenvalues of the linearized evolution equation crosses the imaginary axis in the complex plane. In this situation, centre manifold theory can be used to extract a low dimensional subsystem which completely captures the local bifurcational behaviour. Such essential models enable the onset of divergence and flutter to be analysed relatively simply and the inclusion of non-linear terms permits the global study of post-bifurcational behaviour. The general approach is illustrated by analysis of two mode models of a pipe and of a panel and some important omissions in previous treatments of linear and undamped systems are discussed.     

Flow-induced oscillations of a cantilevered pipe conveying fluid with base excitation

It is known that a plain cantilevered pipe conveying fluid loses its stability by a Hopf bifurcation, leading to either planar or non-planar flutter for flow velocities beyond the critical flow velocity for Hopf bifurcation. If an external mass is attached to the end of the pipe (an end-mass), the resulting dynamics become much richer, showing 2D and 3D quasiperiodic and chaotic oscillations at high flow velocities. In this paper, a cantilevered pipe, with and without an end-mass, subjected to a small-amplitude periodic base excitation is considered. A set of three-dimensional nonlinear equations is used to analyze the pipe?s response at various flow velocities and with different amplitudes and frequencies of base excitation. The nonlinear equations are discretized using the Galerkin technique and the resulting set of equations is solved using Houbolt?s finite difference method. It is shown that for a plain pipe (with no end-mass), non-planar post-instability oscillations can be reduced to planar periodic oscillations for a range of base excitation frequencies and amplitudes. For a pipe with an end-mass, similarly to a plain pipe, three-dimensional period oscillations can be reduced to planar ones. At flow velocities beyond the critical flow velocity for torus instability, the three-dimensional quasiperiodic oscillations can be reduced to two-dimensional quasiperiodic or periodic oscillations, depending on the frequency of base excitation. In all these cases, a low-amplitude base excitation results in reducing the three-dimensional oscillations of the pipe to purely two-dimensional oscillations, over a range of excitation frequencies. These numerical results are in agreement with the previous experimental work.     

周期孔洞区域中热力耦合问题的双尺度有限元计算

复合材料的研究中经常遇到具有周期孔洞结构的材料,由于区域的小周期性及剧烈振荡性,用传统的有限元计算方法来计算这些材料对应的问题时需要大量的计算机存储空间及计算时间.对这类材料的热力耦合问题给出了一种新型的高阶双尺度渐近解,得到了对应的均匀化常数、均匀化方程及对应的有限元算法.数值算例表明,周期单胞的局部结构对局部应力与应变有较大的影响.算法对数值模拟这类材料的力学行为是高效和可行的. 关键词： 双尺度方法 热力耦合 周期孔洞区域 有限元方法     

The prediction of flow-induced noise in heat exchanger tube arrays

A number of methods for the prediction of flow-induced acoustic standing waves in heat exchangers are recommended in the literature. The source for this noise has been assumed to be vortex shedding, turbulent buffeting or broadband turbulence and a variety of methods based on these have been proposed for predicting the occurrence of these standing wave resonances. Furthermore, parameters which estimate the capacity of a heat exchanger unit to dissipate acoustic energy and thus suppress such resonances have been suggested. As there has been no direct comparison of these various techniques, there is confusion as to the applicability of each. In this paper the merits of the techniques for predicting resonance are compared, with use of data from four independent sources, and a method for estimating the limit conditions for avoidance of resonance is recommended. In addition, the parameters for estimating the “damping capacity” of a tube bank are examined and shown to have limitations. A modified damping criterion is suggested and appears to correlate existing data well.     

On the density oscillations in finite systems

We propose an analytic form of the dispersion law for an elementary excitation in infinite nuclear matter that entails the existence of zero energy (and therefore, static) roton-like waves. An expression for the charge density of very large nuclei follows from this hypothesis. The application to the case of ²⁰⁸Pb permits a reasonable account of the available experimental data. The analysis is in agreement with the accepted view that nuclear matter is much less dense than a hard sphere Bose-gas like He II, but dense enough to display significant dispersion in the energy versus momentum curve for an elementary excitation.     

圆柱形腔流激振荡及其耦合共振的研究

本文研究了低流速范围内突出式和陷落式圆柱形腔的流激自持振荡和驻波共振问题。根据大量实验数据,归纳出了适合于低流速情况陷落式和突出式腔的流体动力振荡频率公式,理论上考虑腔口辐射阻抗得到了圆柱腔的驻波共振频率公式,其理论值与试验值符合很好。着重研究了两种振荡的耦合共振,根据本文得出的两种振荡的频率公式,给出了圆柱形腔满足耦合共振条件时腔深和流速之间的关系,实测值与计算值符合很好。     

A tensor artificial viscosity using a finite element approach

We derive a tensor artificial viscosity suitable for use in a 2D or 3D unstructured arbitrary Lagrangian–Eulerian (ALE) hydrodynamics code. This work is similar in nature to that of Campbell and Shashkov [1]; however, our approach is based on a finite element discretization that is fundamentally different from the mimetic finite difference framework. The finite element point of view leads to novel insights as well as improved numerical results. We begin with a generalized tensor version of the Von Neumann–Richtmyer artificial viscosity, then convert it to a variational formulation and apply a Galerkin discretization process using high order Gaussian quadrature to obtain a generalized nodal force term and corresponding zonal heating (or shock entropy) term. This technique is modular and is therefore suitable for coupling to a traditional staggered grid discretization of the momentum and energy conservation laws; however, we motivate the use of such finite element approaches for discretizing each term in the Euler equations. We review the key properties that any artificial viscosity must possess and use these to formulate specific constraints on the total artificial viscosity force term as well as the artificial viscosity coefficient. We also show, that under certain simplifying assumptions, the two-dimensional scheme from [1] can be viewed as an under-integrated version of our finite element method. This equivalence holds on general distorted quadrilateral grids. Finally, we present computational results on some standard shock hydro test problems, as well as some more challenging problems, indicating the advantages of the new approach with respect to symmetry preservation for shock wave propagation over general grids.     

A finite element analysis of the reactive element effect in oxidation of chromia-forming alloys

In many situations, failures such as spallation or cracking occur when chromia-forming alloys are subjected to high-temperature thermal treatment. On the other hand, a small amount of reactive element addition can remarkably increase the adherence between chromia scales and alloys. A two-dimensional finite element model has been developed to analyze the effects of reactive elements on the selective oxidation of chromia scale-forming binary alloys. The quantitative relation between the diffusivities of chromium ions and the activities of reactive element (yttrium) has been derived for the first time and has been incorporated with a continuum thermodynamic model accounting for stress–diffusion interaction in the oxidation of Cr–Fe alloys. The model predicts that the diffusivities of Cr ions are affected by the activities of reactive element compounds, which eventually leads to a decrease of the interfacial spallation driving force, as well as an increase of the interfacial adherence.     

A finite element analysis for the vibration modes of a bladed disc

The coupled vibration modes of a rotating blade-disc system are calculated by a finite element method. It is assumed that a large number of identical blades are present, such that the resulting blade loadings on the disc can be considered continuously distributed around the rim of the disc. The disc may have arbitrary profile, and the blades may be tapered and twisted, thus closely representing practical axial flow turbomachine configurations. The effects of rotation, thermal stress, and transverse shear and rotatory inertia in discs of moderately thick profile are readily incorporated into the finite element model. Calculated values of frequencies are compared with experimental data obtained on non-rotating models, and the convergence of the solution is examined by comparison with exact solutions, which can be obtained for configurations of simple geometry. Excellent agreement with experimental data is obtained when using remarkably few elements in the mathematical model, and convergence of the solution is extremely rapid.     

A finite element model using multi-layered shell element in laser forming

Due to its enormously high flexibility laser forming has been gaining importance in recent years. This rapidness and flexibility demand very precise controlling strategies especially when simulating the process of large plates and challenging the limited computation power of the current workstation. A simple, robust and accurate modeling method of laser forming has been demonstrated to solve this problem. The simplified model is meshed by multi-layered shell element, simulated with a more real scanning method and fewer parameters. The intelligent meshing strategies have reduced the number of elements dramatically. Thus the simulation efficiency has been improved significantly. By comparing the simulation results under the simplified model with the results under the traditional model for laser forming, the applicability of proposed method has been proven. The method of these simplified models is also suitable to simulate complex finite element models, which take much time to simulate. It would throw some light on the thermal mechanically coupled-field simulation of large sheet.     

Acoustic streaming with resonance gas oscillations in a cylindrical tube

Acoustic streaming accompanying acoustic resonance oscillations of gas in a tube is considered. The effect of both the Prandtl number and the wall loss on the velocity of acoustic streaming in a viscous heat-conducting medium is investigated. Expressions for the longitudinal and transverse components of the flow velocity are obtained.     

Synchronization of oscillations in a backward-wave tube: Theory and experiment

Synchronization of oscillations in an electron-wave system with a backward electromagnetic wave is studied theoretically and experimentally. The characteristics of nonautonomous oscillations in the backwardwave tube are analyzed with allowance for the space charge and physical processes accompanying the transition of the distributed autonomous system to the synchronization regime. Theoretical results are compared with experimental data.     

Fallopian tube analysis of the peristaltic-ciliary flow of third grade fluid in a finite narrow tube

The present prospective theoretical investigation deals with analysis of the peristaltic-ciliary transport of a developing embryo within the fallopian tubal fluid in the human fallopian tube. A mathematical model of peristalsis-cilia induced flow of viscoelastic fluid characterized by the third grade fluid model within the fallopian tubal fluid in a finite two dimensional narrow tube is developed. Non-linear partial differential equation resulting from the modelling of the proposed model is solved using perturbation method. Flow variables like axial and radial velocities, appropriate residue time over tube length, pressure difference over wavelength and stream function are analyzed for embedded parameters and constants. Salient features of the pumping characteristics and trapping phenomenon are discussed in detail. The analysis showed that embedded parameters and constants have opposite effects on axial velocity and appropriate residue time over tube length. Moreover, a comparison of the peristaltic flow with the peristaltic-ciliary flow and the third grade fluid with the linearly viscous fluid is made as a special case. The relevance of the current results to the transport of a developing embryo within the fallopian tubal fluid is also explored. It reveals that, third grade fluid instead of the linearly viscous fluid and the inclusion of cilia along with peristalsis help to complete the required mitotic divisions while transporting the developing embryo within the fallopian tubal fluid in the human fallopian tube.     

Resonant oscillations of a gas in an open tube in the shock-free wave mode

Nonlinear gas oscillations excited in an open tube by a flat piston at one of the tube ends are studied. The sinusoidal piston oscillations in the shock-free wave mode are created by a vibration exciter near the first eigenfrequency. Expressions for gas pressure oscillations are obtained for a tube with a nonrounded end without a flange and secondary flow velocity components. The influence of the piston displacement amplitude on the pressure range and secondary flow velocity of gas is studied. The theoretical calculations of the gas pressure are compared with experimental data. An estimate for the velocity of particle motion along the tube axis is presented with calculated values of the secondary flow velocity.     

A finite element scheme to study the nonlinear optical response of a finite grating without and with defect

We present a simple numerical scheme based on the finite element method (FEM) using transparent-influx boundary conditions to study the nonlinear optical response of a finite one-dimensional grating with Kerr medium. Restricting first to the linear case, we improve the standard FEM to get a fourth order accurate scheme maintaining a symmetric-tridiagonal structure of the finite element matrix. For the full nonlinear equation, we implement the improved FEM for the linear part and a standard FEM for the nonlinear part. The resulting nonlinear system of equations is solved using a weighted-averaged fixed-point iterative method combined with a continuation method. To illustrate the method, we study a periodic structure without and with defect and show that the method has no problem with large nonlinear effect. The method is also found to be able to show the optical bistability behavior of the ideal and the defect structure as a function of either the frequency or the intensity of the input light. The bistability of the ideal periodic structure can be obtained by tuning the frequency to a value close to the bottom or top linear band-edge while that of the defect structure can be produced using a frequency near the defect mode or near the bottom of the linear band-edge. The threshold value can be reduced by increasing the number of layer periods. We found that the threshold needed for the defect structure is much lower then that for a strictly periodic structure of the same length.     

A flux tube model of the finite temperature deconfining transition in QCD

A flux tube model describing the finite temperature deconfining transition in QCD is constructed. The difference between SU(2) and SU(3) is pointed out. An explanation for the disappearance of the SU(3) transition for light quark masses is given.     

A finite element model for ultrasonic cutting

Using a single-blade ultrasonic cutting device, a study of ultrasonic cutting of three very different materials is conducted using specimens of cheese, polyurethane foam and epoxy resin. Initial finite element models are created, based on the assumption that the ultrasonic blade causes a crack to propagate in a controlled mode 1 opening, and these are validated against experimental data from three point bend fracture tests and ultrasonic cutting experiments on the materials. Subsequently, the finite element model is developed to represent ultrasonic cutting of a multi-layered material. Materials are chosen whose properties allow a model to be developed that could represent a multi-layer food product or biological structure, to enable ultrasonic cutting systems to be designed for applications both in the field of food processing and surgical procedures. The model incorporates an estimation of the friction condition between the cutting blade and the material to be cut and allows adjustment of the frequency, cutting amplitude and cutting speed.     

A new hybrid cylindrical shell finite element

An assumed stress distribution is used to derive the stiffness matrix for a rectangular cylindrical shell element. A numerical method is given for selecting the required number of terms in the stress assumption. A selection of various static and dynamic results are presented and compared with results obtained by exact theory and other finite elements.