Vibration Analysis of a Transversely Isotropic Hollow Cylinder by Using the Matrix Frobenius Method

The aim of the paper is to study three-dimensional free vibrations in a homogeneous transversely isotropic, thermoelastic hollow cylinder, which is initially undeformed and at uniform temperature. The surfaces of the cylinder are subjected to stress free and thermally insulated or isothermal boundary conditions. The displacement potential functions have been introduced in the equations of motion and heat conduction to decouple purely shear and longitudinal motions. The purely transverse wave is not affected by thermal field. By using the method of separation of variables, the system of governing partial differential equations is reduced to four second-order coupled ordinary differential equations in radial coordinate. The Matrix Frobenius method of the extended power series is employed to obtain the solution in radial direction. To illustrate the analytic results, the numerical solution of various relations and equations has been carried out to compute the lowest frequency, frequency shift and damping factor of vibrations in a hollow cylinder of zinc material with MATLAB software programming. The computer simulated results have been presented graphically.     

Three-dimensional vibration analysis in transversely isotropic cylinder with matrix Frobenius method

Abstract

This article deals with the study of three-dimensional vibrations in stress free as well as rigidly fixed, thermally insulated (or isothermal), homogeneous transversely isotropic solid cylinder under the purview of three-phase lag model of generalized thermoelasticity. The displacement potential functions have been introduced in the equations of motion and heat conduction in order to decouple the purely shear and longitudinal motions. The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. Circumferential wave propagation in cylindrical curved plate is discussed. To illustrate the analytic results, the numerical solution of various relations and equations have been carried out to compute the lowest frequency, inverse quality factor, and dissipation factor of vibrations and the computer-simulated results are presented graphically for different thermoelastic models.     

Thermally Developing Electro-osmotic Convection in Microchannels with Finite Debye-Layer Thickness

ABSTRACT

Thermally developing electro-osmotically generated flow with in circular microtubes with finite Debye-layer thickness has been analyzed. This study focuses on finite Debye-layer effects, a scenario for which the velocity distribution across the tube cross section varies with the ratio of tube radius and Debye length (termed here the relative microtube radius). Numerical solution of the hydrodynamically developed, thermally developing transport for such a flow is presented in this article. The effect of variations in the relative microtube radius and strength of the Joule and viscous heating on the thermal transport are explored over the possible ranges of the governing parameters.     

Numerical simulation of a submerged cylindrical wave energy converter

In this study, a numerical model based on the complete solution of the Navier–Stokes equations is proposed to predict the behavior of the submerged circular cylinder wave energy converter (WEC) subjected to highly nonlinear incident waves. The solution is obtained using a control volume approach in conjunction with the fast-fictitious-domain-method for treating the solid objects. To validate the model, the numerical results are compared with the available analytical and experimental data in various scenarios where good agreements are observed. First, the free vibrations of a solid object in different non-dimensional damping ratios and the free decay of a heaving circular cylinder on the free surface of a still water are simulated. Next, the wave energy absorption efficiency of a circular cylinder WEC calculated from the model is compared with that of the available experiments in similar conditions. The results show that tuning the converter based on the linear theory is not satisfactory when subjected to steep incident waves while the numerical wave tank (NWT) developed in the current study can be effectively employed in order to tune the converter in such conditions. The current NWT is able to predict the wave-body interactions as long as the turbulence phenomena are not important which covers a wide range of Reynolds and Keulegan-Carpenter numbers.     

The effect of viscous dissipation in thermally fully-developed electro-osmotic heat transfer in microchannels

The influence of viscous dissipation on thermally fully-developed, electro-osmotically generated flow has been analyzed for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Such a flow is established not by an imposed pressure gradient, but by a voltage potential gradient along the length of the tube. The result is a combination of unique electro-osmotic velocity profiles and volumetric heating in the fluid due to the imposed voltage gradient. For large ratio of the microtube radius (or microchannel half-width) to Debye length, the wall-normal fluid velocity gradients can be extremely high, which has the potential for significant viscous heating. The solution for the fully-developed, dimensionless temperature profile and corresponding Nusselt number have been determined for both geometries and for both thermal boundary conditions. It is shown that three dimensionless parameters govern the thermal transport: the relative duct radius (ratio of the duct radius or plate gap half-width to Debye length), the dimensionless volumetric source (ratio of Joule heating to wall heat flux), and a dimensionless parameter that relates the magnitude of the viscous heating to the Joule heating. Surprisingly, it is shown that the influence of viscous dissipation is only important at low values of the relative duct radius. For magnitudes of the dimensionless parameters which characterize most practical electro-osmotic flow applications, the effect of viscous dissipation is negligible.     

CFD based evaluation of heat transfer coefficient from cylindrical particles

Two- and three-dimensional CFD modeling of heat transfer from discrete circular cylindrical particles in four different situations including A) infinite cylinder in cross-flow, B) cross-flow on finite cylinder with different aspect ratio in a rectangular duct, C) axial-flow on finite cylinder and D) axial-flow on finite cylinder with upstream turbulence have been investigated with the commercial CFD software, FEMLAB. The results were validated using experimental data from different research papers and also experimental correlations and show good quantitative and qualitative agreement with each other. In case B, a correction term has been proposed from CFD work which is applied to an experimental correlation to consider the aspect ratio influence on predicting the Nusselt number with an average error of 3.7%.     

THREE-DIMENSIONAL TRANSIENT THERMAL STRESSES IN A FINITE CIRCULAR CYLINDER UNDER NONAXISYMMETRIC TEMPERATURE DISTRIBUTION

A general analysis is developed for the complete three-dimensional transient thermal-stress distribution in a finite circular cylinder subject to arbitrary heating of the lateral surface with arbitrary heat transfer into the surrounding medium. The problem is formulated in terms of the modified Boussinesq functions and the thermoelastic displacement potential, which seem convenient in form and have the advantage of being applicable to non-axisymmetric problems. The solution meets the tractionfree boundary conditions on both the lateral surface and the plane ends of the cylinder. Graphical illustrations are provided.     

EFFECTS OF THERMAL BOUNDARY CONDITION ON BUOYANCY DRIVEN TRANSITIONAL AIR FLOW IN A VERTICAL CYLINDER HEATED FROM BELOW

A three-dimensional unsteady numerical computation was carried out here to investigate the effects of the thermal boundary condition on the convection flow in a vertical, bottom heated cylinder containing air. The thermal condition at the sidewall of the cylinder is assumed to be thermally well insulated or perfectly conducting. Results were obtained for air in a cylinder of finite aspect ratio (Gamma = 2) for various thermal Rayleigh numbers. The predicted results indicate that the flow in the sidewall insulated cylinder is highly asymmetric even at steady state and contains multicellular vortices. The flow formation processes leading to the above structures are relatively complicated. In the early transient two axisymmetric circular vortex rolls, one on top of another, appear. Then the rolls merge asymmetrically. In the late stage the flow deflection by the cylinder top and bottom results in a very complex flow. In the cylinder with a perfectly conducting sidewall the transition from a steady to a time dependent flow is subcritical. However, in the cylinder with an insulated sidewall the flow transition is supercritical.     

大尺度展向波形圆柱绕流流场结构的数值研究

针对大尺度展向波形圆柱绕流的减阻特性,通过大涡模拟（LES）研究波形振幅对圆柱体绕流流场结构的影响,获得波形圆柱体绕流气动性能曲线、尾迹时均流速分布和非定常涡量场分布,最后与直圆柱绕流的流场结构进行对比分析。结果表明,波形圆柱绕流的平均阻力系数小于直圆柱体绕流,流向涡的形成改变了圆柱近尾迹区的流场结构,因此,波形圆柱体尾迹涡系表现得更为紧凑,尾迹涡流得到拉伸与破裂。在亚临界雷诺数为3000时,最大阻力系数减少18.3%,最优振幅比为0.152;且波形圆柱体的升力波动大大减少,甚至得到抑制。由于波形表面会形成更稳定的三维自由剪切层,这样的自由剪切层在下游位置卷起漩涡,大大地改变了圆柱周围的流场结构。研究表明振幅比在确定波形圆柱后面的三维涡旋结构中起着至关重要的作用,并对升力波动和流动阻力的降低有着显著的影响。     

Numerical investigation of laminar natural convective heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure

A numerical simulation was conducted to investigate the steady laminar natural convective heat transfer for air within the horizontal annulus between a heated triangular cylinder and its circular cylindrical enclosure. The Boussinesq approximation was applied to model the buoyancy-driven effect and the governing equations were solved using the finite volume method. Four different Rayleigh numbers and four different radius ratios were considered, and four different inclination angles for the inner triangular cylinder were investigated as well. The computed flow and temperature fields were demonstrated in the form of streamlines and isotherms. Variations of the maximum stream function and the local and average Nusselt numbers were displayed as functions of the above-mentioned parameters. Correlations of the average Nusselt number were proposed based on curve fitting. At constant radius ratio, inclination angles of the inner triangular cylinder are found to have negligible effects on the average Nusselt number.     

Fully developed electro-osmotic heat transfer in microchannels

Thermally fully developed, electro-osmotically generated convective transport has been analyzed for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Such a flow is established not by an imposed pressure gradient, but by a voltage potential gradient along the length of the tube. The result is a combination of unique electro-osmotic velocity profiles and volumetric heating in the fluid due to the imposed voltage gradient. The exact solution for the fully developed, dimensionless temperature profile and corresponding Nusselt number have been determined analytically for both geometries and both thermal boundary conditions. The fully developed temperature profiles and Nusselt number are found to depend on the relative duct radius (ratio of the Debye length to duct radius or plate gap half-width) and the magnitude of the dimensionless volumetric source.     

Three-dimensional analyses of surface cracks in pressurised thick-walled cylinders

Stress intensity factors for semi-elliptical surface cracks in internally pressurised thick-walled cylinders of radius ratio 3 are presented for a wide range of crack sizes. These solutions were obtained using the boundary integral equation method for three-dimensional stress analysis. Only one crack shape is considered—a semi-ellipse with the length of its semi-minor axis equal to 0·6 times the length of its semi-major axis —but the ratio of crack depth to wall thickness ranged from 0·2 to 0·8. Hoop strain distributions at the outer circumference of the cylinder are also presented for the different crack sizes analysed; the results are useful for experimentally monitoring crack growth.     

Effect of Prandtl Number on Free Convection from Two Cylinders in a Square Enclosure

This study explores the effect of Prandtl number on the laminar natural convection heat transfer to Newtonian fluids in a square enclosure consisting of one hot circular cylinder and one cold circular cylinder. The walls of the square enclosure are maintained isothermal and at the same temperature as the cold cylinder and the fluid medium. The governing partial differential equations have been solved numerically over the following ranges of conditions: Grashof number, 10 to 10⁵; Prandtl number, 0.7 to 100 (or the range of Rayleigh numbers as 7 to 10⁷); and relative positioning of the cylinders, ?0.25 to 0.25. However, the ratio of the radius of the cylinder to the side of the enclosure is held fixed at 0.2. Extensive results on the streamline and isotherm contours, the local Nusselt number distribution, and the average Nusselt number are discussed to delineate the influence of Grashof and Prandtl numbers on them for a given location with respect to the horizontal center line. The surface-averaged Nusselt number shows a positive dependence on Grashof and Prandtl numbers for a fixed location of the two cylinders. The heat transfer results have been correlated as a function of the Rayleigh number and geometric parameters, thereby enabling its prediction in a new application.     

THERMAL STRESSES IN A LONG ELASTIC CYLINDER CONTAINING A PENNY-SHAPED CRACK AND EMBEDDED IN A THERMALLY CONDUCTIVE ELASTIC INFINITE MEDIUM

This paper is concerned with- the state of stress in a long circular elastic cylinder with a concentric penny-shaped crack whose surfaces are subjected to a prescribed temperature. The plane of the crack is assumed to be perpendicular to the axis of the cylinder. The cylinder is bonded to a thermally conductive elastic infinite medium. The thermal and elastic constants of the cylinder and infinite medium are assumed to be different. By assuming suitable representations for the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Similarly, the thermoelastic problem is also reduced to the solution of a Fredholm integral equation of the second kind. A closed-form expression is obtained for the stress-intensity factor. The integral equations are solved numerically, and the results are used to obtain numerical values for the stress intensity factor. These values are presented graphically.     

Universal SCFs and optimal chamfering in cross-bored cylinders

A three-dimensional finite element computer program was developed to establish the stress distributions and stress concentration factors (SCFs) in chamfered cross-bored cylinders under internal pressure. The displacement formulation using eight noded brick and four noded tetrahedron isoparametric elements was used. The Frontal solution technique was used due to limited computing facilities. For several thickness ratios and cross bore to main bore radius ratios, the variation of SCF with chamfer angle for various chamfer length (clr) ratios was investigated. In each case, a universal SCF value corresponding to a unique value of chamfer angle was established. For most clr and chamfer angle combinations, the SCF was higher than that of a plain cross-bored cylinder. However, for some combinations, the SCF curve had an optimum value lower than that of a plain cross-bored cylinder. In optimal chamfered cylinders with thickness ratio between 2.25 and 3, the SCF was found to increase with decrease of thickness ratio. Where thickness ratio was between 1.25 and 2, a cross bore to main bore radius ratio of 0.075 was found to be a geometrical constant with a corresponding SCF of 2.65. Thick cylinders were found to be more suited to chamfering than thin cylinders. The resulting data in this work provides a useful and quick design tool.     

Fractional Order Thermoelastic Problem for an Infinitely Long Solid Circular Cylinder

In this study, we consider an infinitely long solid circular cylinder, whose lateral surface is traction free and subject to a known surrounding temperature. The problem is in the context of the fractional order thermoelasticity theory. The medium is assumed to be initially quiescent. The solution is obtained by a direct approach without the customary use of potential functions. Laplace transform technique is used to obtain the general solution for any set of boundary conditions. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical results are given and represented graphically.     

Free Convection Heat Transfer from a Confined Horizontal Elliptic Cylinder

The laminar free convection heat transfer from an isothermal horizontal cylinder of elliptical cross-section confined between two adiabatic walls is investigated by the Mach-Zehnder interferometry technique. The ellipse major axis is vertical, and the minor to major axis ratio is kept constant to 0.53. This paper focuses on the effect of wall spacing and Rayleigh number variation on the local and average free convection heat transfer coefficient from the cylinder surface. The local and average Nusselt numbers were determined for the Rayleigh number range of 9 × 10 ² to 3.2 × 10 ³ and wall spacing to cylinder minor axis ratios of 1.9, 2.3, 2.67, 3.17, 3.8, 4.6, 6.12, 8, 13, ∞. Results are indicated with a single correlation that gives the average Nusselt number as a function of the ratio of the wall spacing to cylinder minor axis and the Rayleigh number. There is an optimum distance between the walls in which the Nusselt number is maximum. The experiment was also carried out on a cylinder of circular cross-section with the same periphery and length of the elliptic cylinder to allow a comparison with the results of other research.     

Numerical study of laminar free convection about a horizontal cylinder with longitudinal fins of finite thickness

Conjugate numerical solution of laminar free convection about a horizontal cylinder with external longitudinal fins of finite thickness has been carried out. Fins alone contribute very small to the total heat transfer but they greatly influence the heat transfer from the uncovered area of the cylinder. Among the various fin parameters, thickness has the greatest influence on heat transfer. The rate of heat transfer is above that for the free cylinder only when the attached fins are very thin. For thin fins, there exist a fin length, which maximizes the rate of heat transfer. The optimum number and dimensionless length of the fins were obtained as 6 and 0.2 respectively when fin thickness is 0.01, the thinnest among those investigated in this study.     

Free Convection Heat Transfer from Vertical Slender Cylinders: A Review

The effect of transversal curvature of a vertical cylinder becomes important where the thermal boundary layer thickness is comparable or thicker then the radius of cylinder. The cylinder slenderness criterion for laminar free convection for fluids of Prandtl numbers from 0.01 to 100 is presented. The classical analysis of the laminar free convection heat transfer from vertical cylinders is shown. Some results of numerical calculations of the turbulent boundary layer on a vertical cylinder using modified integral method are given. Experimental data concerning the laminar-turbulent transition suggest that the critical Grashof number for a vertical flat plate is Gr _cr ≈ 10⁹ and for a vertical cylinder is Gr _cr ≈ 4 × 10⁹. Theoretical, numerical, and experimental data for free convection heat transfer from vertical slender circular cylinders are surveyed. A separate section of the paper is devoted to the presentation of the list of selected correlation equations. Some of them are compared graphically. In the laminar region, the correlation equation based on the numerical calculations is validated with the recent experimental results for Prandtl number of 0.71 and for the cylinder height to diameter ratio from 1 to 60. In the turbulent region, few experimental data are available, and some results indicate that the effect of transversal curvature on the average convective heat transfer is very weak.     

Three-Dimensional Pyroelectric Analysis of a Functionally Graded Piezoelectric Hollow Sphere

A three-dimensional analytical piezothermoelastic solution is presented for a functionally graded piezoelectric spherical shell subjected to various thermal boundary conditions applied on the inner and outer surfaces. Material properties are assumed to vary along the radius, r, obeying a power law. Both the thermal field and the pyroelectric responses are resolved using the state space method. On introducing three displacement and two stress functions, two independent kinds of state equations are derived from the basic equations of piezoelectricity. It is interesting that the first kind is a homogeneous equation only related to the purely elastic behavior of the sphere, yet the second has an inhomogeneous term associated with the thermal effect to determine the pyroelectric responses. The state equations are solved by expanding the field variables into series of spherical harmonic functions. Numerical examples are performed to investigate the influence of material inhomogeneity on the pyroelectric responses of the spherical shell.