Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials

Summary The axisymmetric problem of a functionally graded, transversely isotropic, annular plate subject to a uniform transverse load is considered. A direct displacement method is developed that the non-zero displacement components are expressed in terms of suitable combinations of power and logarithmic functions of r, the radial coordinate, with coefficients being undetermined functions of z, the axial coordinate. The governing equations as well as the corresponding boundary conditions for the undetermined functions are deduced from the equilibrium equations and the boundary conditions of the annular plate, respectively. Through a step-by-step integration scheme along with the consideration of boundary conditions at the upper and lower surfaces, the z-dependent functions are determined in explicit form, and certain integral constants are then determined completely from the remaining boundary conditions. Thus, analytical elasticity solutions for the plate with different cylindrical boundary conditions are presented. As a promising feature, the developed method is applicable when the five material constants of a transversely isotropic material vary along the thickness arbitrarily and independently. A numerical example is finally given to show the effect of the material inhomogeneity on the elastic field in the annular plate.     

Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders

We use the Airy stress function to derive exact solutions for plane strain deformations of a functionally graded (FG) hollow cylinder with the inner and the outer surfaces subjected to different boundary conditions, and the cylinder composed of an isotropic and incompressible linear elastic material. For the shear modulus given by either a power law or an exponential function of the radius r, we derive explicit expressions for stresses, the hydrostatic pressure and displacements. Conversely, we find the variation with r of the shear modulus for a linear combination of the radial and the hoop stresses to have a pre-assigned variation in the cylinder; this inverse problem is usually called material tailoring. The shear modulus found while solving the inverse problem must be positive everywhere. Results for a few problems are computed and presented graphically. It seems that the Airy stress function approach is used here for the first time to analyze two-dimensional problems for incompressible materials. When studying axisymmetric deformations of an FG cylinder, it is found that for the hoop stress to be uniform through the cylinder thickness the shear modulus must be proportional to the radial coordinate r as found earlier by Batra [Batra RC. Optimal design of functionally graded incompressible linear elastic cylinders and spheres. AIAAJ 2008;46(8):2005–7.] and for the maximum in-plane shear stress to be constant the shear modulus must vary as r². The expression for the maximum in-plane shear stress in terms of pressures and the radii of the inner and the outer surfaces of the cylinder is a universal result valid for all materials for which the shear modulus is proportional to r². For a hollow cylinder fixed on the inner surface and subjected to tangential tractions on the outer surface (or vice versa) the through-the-thickness in-plane shear stress distribution is also universal and is determined by surface tractions and the outer radius of the cylinder; it is independent of the spatial variation of the shear modulus.     

Magnetoelastodynamic stress and perturbation of magnetic field vector in an orthotropic laminated hollow cylinder

An analytical method is presented to solve magneto-elastic wave propagation and perturbation of the magnetic field vector in an orthotropic laminated hollow cylinder with arbitrary thickness. The magnetoelastodynamic equation for each separate orthotropic hollow cylinder is solved by making use of finite Hankel transforms and Laplace transforms. The unknown constants involved in the solution for each separate layer are determined by using the interface continuity conditions between layers and the boundary conditions at the internal and external boundaries of the laminated hollow cylinders. Thus, an exact solution for magnetoelastodynamic stresses and perturbation response of an axial magnetic field vector in laminated hollow cylinder is obtained. From sample numerical calculations, it is seen that the present method is suitable to solve magnetoelastodynamic problems of laminated hollow cylinders subjected to a radial shock load and an axial magnetic field.     

A boundary layer theory for the end problem of a circular cylinder

Summary This paper discusses the nature of an approximate solution for the hollow circular cylinder whose fixed ends are given a uniform relative axial displacement and whose cylindrical surfaces are free from traction. We shall take the solution of this problem to be given by a super-position of the following two problems: problem I considers a finite length cylinder whose ends are given a relative axial displacement, but are no longer fixed; problem II removes the radial displacement at the end of the cylinder obtained in problem I.Nomenclature a mid-surface radius of cylinder - c half-height of cylinder - E, in-plane elastic moduli - E_t, _t, G_t transverse elastic moduli - _z, , _r axial, circumferential, and normal strain - _rz transverse shear strain - h cylinder thickness - _z, , _r axial, circumferential, and normal stress - _rz transverse shear stress - z, r axial and radial coordinates - u_z, u_r axial and normal displacements     

Maximum density effects on natural convection in a vertical annulus filled with a non-Darcy porous medium

Summary This article numerically studies the problem of natural convection in a porous medium saturated with cold water, under density inversion, within a vertical annulus. In modeling the flow in the porous medium the non-Darcy effects, which include the Forchheimer inertia and Brinkman viscous effects are taken into account. The governing equations are solved numerically by the finite difference method using the modified strongly implicit procedure. The effects of the inversion parameter _m , radius ratioR ^*, aspect ratioAR, Forchheimer inertia parameter Fc/Pr, and Darcy number parameter Da on the heat transfer and fluid flow characteristics are discussed in detail. Results show that both the inversion parameter and radius ratio have a significant influence on the flow structure and heat transfer rate in the annulus. It is also found that the mean Nusselt number decreases as the Forchheimer inertia parameter or the Darcy number increases. Moreover, the results obtained here are also compared and favorably agree with numerical results and with experimental data.Nomenclature AR aspect ratio,H/L - d particle diameter - Da Darcy number,K/(L ²) - Fc Forchheimer number,K/L - g gravitational acceleration - H annulus height - K permeability - K transport property defined in Eq. (5) - L gap width,r _o –r _i - Nu _i , Nu _o local Nusselt number of the inner and outer cylinders, respectively - mean Nusselt number of the inner and outer cylinders, respectively - p pressure - Pr Prandtl number, / - q constant in Eq. (9) - r radial coordinate - R dimensionless radial coordinate,(r–r _i )/L - r _i radius of inner cylinder - r _o radius of outer cylinder - R ^* radius ratio,(r _o –r _i )/r _i - Ra Rayleigh number,K _m gL(T _h –T _c ) ^q / - T dimensional temperature - T _c dimensional temperature of inner cylinder - T _h dimensional temperature of outer cylinder - T _m temperature corresponding to the density maximum, 4.029325°C - u, v Darcian velocity components inr andz directions, respectively     

Thermal shock analysis and thermo-elastic stress waves in functionally graded thick hollow cylinders using analytical method

Transient stress field and thermo-elastic stress wave propagation are studied in functionally graded thick hollow cylinder under arbitrary thermo-mechanical shock loading, in this article. Thermo-mechanical properties of functionally graded (FG) cylinder are assumed to be temperature independent and vary continuously and smoothly in the radial direction. The governing dynamic equations are analytically solved in temperature and elastic fields. To solve the problem, Laplace transform is used respect to time in all constitutive equations and boundary conditions. At first, temperature field equation analytically solved using Laplace transform and series method. The dynamic behaviors of thermo-elastic stresses are illustrated and discussed for various grading patterns of thermo-mechanical properties in several points across the thickness of FG cylinder. Time history of temperature field and thermal stresses are obtained using the residual theorem and the fast Laplace inverse transform method (FLIT), respectively. Also, the effects of the cylinder thickness and convection heat transfer coefficient on dynamic response of FG cylinder are revealed and discussed. The presented analytical method provides a ground to study the time histories of radial and hoop stresses in FG cylinders with different thickness and various volume fraction exponents. The advantage of this method is its mathematical ability to support simple and complicated mathematical function for the thermo-mechanical boundary conditions. A reasonable agreement can be seen in comparison of obtained results based on the presented analytical method with published data.     

On the elastic deformation of non-saturated swelling soils

Summary This paper aims at completing and extending the theories as they have been applied to swelling media for the last 50 years, to swelling non-saturated soils. However, having regard to the complicated behaviour of swelling soils, it was thought necessary to keep the state of stress as simple as possible when discussing swelling in cylindrical specimens in which drainage is completely prevented. Definitions of the parameters are attemptedbased on equilibrium thermodynamics. Contributions to swelling stress calculation, when expansion is considered in relation to vapor pressure and moisture content, are also given.Notation u _w hydrostatic pressure of soil water - total stress - effective stress - m _c soil mass - m _w water mass - H=(m _w/m _c) moisture content (gravimetric) - A=(m _w/m _c)_Xi (i=x,y,z orr, ,z) moisture content at constant swelling pressure (gravimetric) - n=(m _w/m _c)_h moisture content at constant vapor pressure (gravimetric) - n _a amount of a constituant phase - h _v vapor pressure of soil water - h ₀ vapor pressure of pure water - h=(h _v/h ₀) relative vapor pressures - M molecular weight of water - specific volume of the soil water vapor - R gas constant (8.3144 Joules/mole·^oK) - T absolute temperature - P swelling pressure of an isotropic soil swelling without constrain - S entropy of a sample of volumeV - O oncotic energy - O _r residual oncotic energy - U internal energy - F Helmholtz free energy - G Gibb's free energy - partial molar Gibb's free energy of constituenta - chemical potential which equals the partial molar Gibb's free energy of constituenta - _v chemical potential of water vapor - _v chemical potential of soil water - V volume - V _H volume at constant moisture content - W _a specific energy for the adsorbed water - W _a0 specific osmotic energy of adsorbed solutes - W ₀ specific osmotic energy of free solutes - x, y, z Cartesian coordinate system - r, ,z cylindrical coordinate system - u, v, z _w displacement components along the strain variablesx, y, andz - z _w height of the water column - P _i (i=x, y, z) loads along the strain variablesx, y, z - P _i (i=x, y, z) orr, ,z mechanical pressure along the strain variables - X _i (i=x, y, z orr, ,z) swelling pressures along the strain variablesx, y, z - P _k hydrostatic swelling pressure defined byp _k=(X _x+X _y+X _z)/3 - P _{k, H} hydrostatic swelling pressure under constant moisture content - l _i (i=x, y, z) extential displacements along the strain variablesx, y, z - s _i ^* (i=x, y, z orr, ,z) differential swelling - s _i (i=x, y, z orx, ,z) differential swelling per unit volume - s _k hydrostatic differential swelling defined bys _k=(s _x+s _y+s _z)/3 - s _h differential swelling under constant vapor pressure - Q effective swelling pressure defined as the energy encompassing all the unknown effects contributing to swelling against the mechanical pressure and the vapor pressure of soil water per unit volume - Q coefficient of swelling - E _i (i=r, ,z orx, y, z) Young's moduli in the directioni - E _{i, h} (i=x, y, z) Young's moduli at constant vapor pressure - B _h Bulk modulus under constant vapor pressure - G _{ij. h} (i, j=x, y, z) Shear modulus under constant vapor pressure - v _ij (i, j=r, ,z orx, y, z) Poisson's coefficient which characterizes the compression in the directioni for tension in the directionj, etc. - e _i (i=r, ,z) final radial, tangential and axial strains - e _{i, s} (i=r, ,z) radial, tangential and axial strains due to swelling - loads along the strain variablesr, ,z - G _ij (i, j=x, y, z) rigidity modulus - g acceleration due to gravity - a radius of the cylinder - r ₁,r ₂ roots of the characteristic equation - W work done by the surroundings on the system     

Static deformations of functionally graded polar-orthotropic cylinders with elliptical inner and circular outer surfaces

Functionally graded materials (FGMs) enable one to tailor the spatial variation of material properties so as to fully use the material everywhere. For example, in a hollow circular cylinder one can vary, in the radial direction, the material moduli to make the hoop stress constant. Whereas the problem for a hollow cylinder with the inner and the outer surfaces circular has been studied, that of a cylinder with a circular outer surface and a non-circular inner surface or vice versa has not been investigated. We study here such a plane-strain problem when the cylinder material is polar-orthotropic, material properties vary exponentially in the radial direction, and deformations are independent of the axial coordinate. The problem is challenging since the cylinder thickness varies with the angular position of a point, and the cylinder material is inhomogeneous. Equilibrium equations are solved by expanding the radial and the circumferential displacements in Fourier series in the angular coordinate. The method of Frobenius series is used to solve ordinary differential equations for coefficients of the Fourier series, and boundary conditions are satisfied in the sense of Fourier series. A parametric study has been conducted that delineates effects on stresses of the eccentricity of the ellipse, the material property gradation index and loads applied on boundaries of the cylinder. The analytical solutions presented here will serve as benchmarks for comparing solutions derived by numerical methods.     

General analytical solution for elastic radial wave propagation and dynamic analysis of functionally graded thick hollow cylinders subjected to impact loading

An analytical method is proposed for the dynamic response analysis of functionally graded thick hollow cylinders under impact loading. The wave motion equation is solved using an analytical method that is based on the composition of Bessel functions. The mechanical properties are considered as power functions of the radius across the thickness of FG cylinder. The FG cylinder is excited by an impact loading at the inner surface of the cylinder, and the plane strain and axisymmetry conditions are assumed for the problem. The time histories of radial displacement and radial and hoop stresses are presented. Also the dynamic response of the FG cylinder is obtained and discussed for various kinds of power function exponents.     

Reliability of stress field in Al–Al2O3 functionally graded thick hollow cylinder subjected to sudden unloading,considering uncertain mechanical properties

In this paper, the reliability analysis and safety evaluation of dynamic stresses are presented for Al–Al₂O₃ functionally graded (FG) thick hollow cylinder subjected to sudden unloading as a mechanical shock loading. The FG cylinder is considered to have infinite length and axisymmetry conditions. The constitutive mechanical properties of Al and Al₂O₃ are assumed as random variables with Gaussian distribution and also the mechanical properties are considered to vary across thickness of FG cylinder as a non-linear power function of radius. The radial and hoop stresses are obtained by solving Navier equation in displacement form and stress–displacement equations. The FG cylinder is divided to many linear functionally graded elements across thickness of cylinder and hybrid numerical method (Galerkin finite element and Newmark finite difference methods) along with the Monte Carlo simulation are employed to solve the statistical problem. The reliability of radial and hoop stresses are calculated in various points across thickness for different grading patterns in functionally graded material (FGM) and several yield stresses. The variability of the dynamic stress reliability of the FG cylinder to the values of coefficients of variation (COVs) is examined and discussed in details.     

Analytical solutions describing the consolidation of a multi-layered soil under circular loading

Analytical solutions describing the consolidation of a multi-layered soil under circular loading are presented. From the governing equations of saturated poroelastic soil in a cylindrical coordinate system, the eighth-order state-space equation of consolidation is obtained by eliminating the variation of time t using the Laplace transform together with the technique of Fourier expansions with respect to the coordinate θ and the Hankel transform with respect to coordinate r. The solution of the eighth-order state-space equation is derived directly by using the Laplace transform and its inversion of the z-domain. Based on the continuity between layers and the boundary conditions, the transfer-matrix method is utilized to derive the solutions for the consolidation of a multi-layered soil under circular loading in the transformed domain. By the inversion of the Laplace transform and the Hankel transform, the analytical solutions in the physical domain are obtained. A numerical analysis based on the solutions is carried out by a corresponding program.     

Strain gradient elasticity solution for functionally graded micro-cylinders

In this paper, strain gradient elasticity formulation for analysis of FG (functionally graded) micro-cylinders is presented. The material properties are assumed to obey a power law in radial direction. The governing differential equation is derived as a fourth order ODE. A power series solution for stresses and displacements in FG micro-cylinders subjected to internal and external pressures is obtained. Numerical examples are presented to study the effect of the characteristic length parameter and FG power index on the displacement field and stress distribution in FG cylinders. It is observed that the characteristic length parameter has a considerable effect on the stress distribution of FG micro-cylinders. Also, increasing material length parameter leads to decrease of the maximum radial and tangential stresses in the cylinder. Furthermore, it is shown that the FG power index has a significant effect on the maximum radial and tangential stresses.     

Stressed state of a hollow two-layer cylinder under dynamic loads

The dynamic problem of an elastic two-layer hollow cylinder whose surfaces are loaded by arbitrary forces is solved by the method based on the use of finite differences solely with respect to time. The numerical calculations of the stress concentration on the inner and outer surfaces and the surface of conjugation as functions of times are carried out for cylinders with different thicknesses of the layers under impact loads. The dependences of stresses on the radial variable r and time parameter τ are illustrated for two-layer cylinders subjected to impacts on the inner and outer surfaces.     

Buckling of composite circular plates under radial compression

Summary Axisymmetric nonlinear buckling equations are introduced for thin elastic circular plates composed of isotropic or orthotropic layers, uniformly compressed in the radial direction.The linear eigenvalue problem is solved in closed form for heterogeneous isotropic circular plates with clamped or simply supported circumference. Invariant buckling parameters are obtained which enable optimization. Several numerical examples are shown.

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Beulen von Kreisplatten aus Verbundwerkstoff unter radialem Druck

Zusammenfassung Für dünne, elastische, in radialer Richtung gedrückte Kreisplatten aus isotropen oder orthotropen Schichten werden drehsymmetrische nichtlineare Beulgleichungen hergeleitet.Das lineare Eigenwertproblem wird für am Rand eingespannte oder frei aufliegende heterogene isotrope Kreisplatten in geschlossener Form gelöst. Invariante Beulparameter, die eine Optimierung ermöglichen, werden erhalten und einige numerische Beispiele angegeben.

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Nomenclature a radius of plate - A _ij elastic area - A _ij ^* extensional rigidity - B _ij elastic statical moment - B _ij ^* , C _ij ^* extensional-flexural coupled rigidities - D _ij elastic moment of inertia - D _ij ^* modified flexural rigidity - flexural rigidity defined in equation (3.32.3) - E Young's modulus - h plate thickness - h ₀ distance to reference plane defined in equation (3.11) - J _i Bessel function of first kind and orderi - L _ij functional operator - m coefficient defined in equation (3.15) - M _r, M radial and circumferential couples, respectively - n constant defined in equation (3.21) - N _r, N radial and circumferential forces, respectively - N ₀ radial compression - p, q coefficients defined in equations (3.32) - Q transverse shear resultant - r radial coordinate - t invariant parameter defined in equation (3.44) - u, w radial and transverse displacements, respectively - x variable defined in equation (3.43) - Y ₁ Bessel function of second kind and first order - z vertical coordinate - change of slope in radial direction - _r0, ₀ radial and circumferential strain components, respectively atz=0 - circumferential coordinate - _r, curvatures in radial and circumferential directions, respectively - v Poisson's ratio - , slope angle before and after deformation, respectively - stress resultant function With 1 FigureThis study was supported by the Israel Academy of Sciences and Humanities and the Technion-Israel Institute of Technology. The numerical examples were carried out at the Technion's Computation Center by I. Smolash.Dedicated to the memory of Mr.Israel Stavsky, father of senior author.Visiting Professor, Washington University, St. Louis, Missouri., 1969/70.     

Similarity analysis of axisymmetric free convection on a horizontal infinite plate subjected to a mixed thermal boundary condition

Summary Similarity analysis of the problem of axisymmetric free convection on a horizontal infinite plate is considered assuming that the plate is subjected to a mixed thermal boundary condition. It is shown that the thermal boundary condition is characterized by a nonnegative parameterm and the two cases ofm=0 andm=1 correspond to prescribed plate temperature and prescribed surface heat flux respectively. If one has to compute the heat transfer coefficient for various values ofm, there is no need to solve the boundary value problem everytime; it is enough to solve a certain polynomial equation provided the solution is known for any particular value ofm.Notation a ₀(r),a ₁(r),a ₂(r) coefficients in Eq. (1) - A transition parameter used in Eq. (22) - C function ofr defined in Eq. (8.1) - f dimensionless stream function - F dimensionless pressure - g acceleration due to gravity - G function ofr defined in Eq. (8.2) - m mixed thermal boundary condition parameter - N normalized heat transfer coefficient in Eq. (26) - Nu _r Nusselt number - p fluid pressure - p _e ambient pressure - P normalized pressure drop at the plate - Pr Prandtl number - r radial coordinate - S normalized stress at the plate - T temperature - T _e ambient temperature - u velocity in the radial direction - w velocity in the axial direction - z axial coordinate - coefficient of thermal expansion - dimensionless similarity variable - dimensionless temperature - exponent inC - kinematic viscosity - ambient fluid density - stream function     

Numerical study of disk driven rotating MHD flow of a liquid metal in a cylindrical enclosure

Summary A numerical study of the steady laminar MHD flow driven by a rotating disk at the top of a cylinder filled with a liquid metal is presented. The governing equations in cylindrical coordinates are solved by a finite volume method. The effect of an axial magnetic field on the flow is investigated for an aspect ratioH/R equal to 1. The magnetic Reynolds number is assumed to be small whereas the interaction parameter,N, is large compared to unity. This allows to derive asymptotic results for the flow solution which are found in good agreement with the numerical calculations. The effect of the top, botton and vertical walls conductivity on the flow is studied. Various combinations of these conductivities are considered. The results obtained showed that one can control the primary flow through a good choice of the electrical conductivity of both the disk and cylinder walls.Notation B Magnetic field - H Height of the cylinder - Ha Hartmann number - jz Axial electric current - N Interaction parameter - P Dimensionless pressure - R Radius of the cylinder - Re Reynolds number - R _m Magnetic Reynolds number - r Dimensionless radius - V _r Dimensionless radial velocity - V _z Dimensionless axial velocity - V Dimensionless azimuthal velocity - Z Dimensionless height Greek symbols Density of the fluid - v Kinematic viscosity - Dynamic viscosity - Electrical conductivity - Angular velocity - Dimensionless electric potential - Thickness of the Ekman layer - Laplacian operator - r Increment of the grid in the radial direction - Z Increment of the grid in the axial direction     

Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method

In this paper, dynamic analysis of nanocomposite cylinders reinforced by single-walled carbon nanotubes (SWCNTs) subjected to an impact load was carried out by a mesh-free method. Free vibration and stress wave propagation analysis of carbon nanotube reinforced composite (CNTRC) cylinders are presented. In this simulation, an axisymmetric model is used. Four types of distributions of the aligned carbon nanotubes (CNTs) are considered; uniform and three kinds of functionally graded (FG) distributions along the radial direction of cylinder. Material properties are estimated by a micro mechanical model. In the mesh-free analysis, moving least squares (MLSs) shape functions are used for approximation of displacement field in the weak form of motion equation and the transformation method was used for the imposition of essential boundary conditions. Effects of the kind of distribution and volume fractions of carbon nanotubes and cylinder thickness on the natural frequencies and stress wave propagation of CNTRC cylinders are investigated. Results obtained for this analysis were compared with FEM and previous published work and good agreement was seen between them.     

Material tailoring for functionally graded hollow cylinders and spheres

We present a technique to tailor materials for functionally graded (FG) linear elastic hollow cylinders and spheres to attain through-the-thickness either a constant hoop (or circumferential) stress or a constant in-plane shear stress. The volume fractions of two phases of a FG material (FGM) are assumed to vary only with the radius and the effective material properties are estimated by using either the rule of mixtures or the Mori-Tanaka scheme; the analysis is applicable to other homogenization methods. For a FG cylinder we find the required radial variation of the volume fractions of constituents to make a linear combination of the radial and the hoop stresses uniform throughout the thickness. The through-the-thickness uniformity of the hoop stress automatically eliminates the stress concentration near the inner surface of a very thick cylinder. The through-the-thickness variations of Young’s moduli obtained with and without considering the variation of Poisson’s ratio are very close to each other for a moderately thick hollow cylinder but are quite different in a very thick hollow cylinder. For an FG sphere the required radial variation of the volume fractions of the two phases to get a constant circumferential stress is similar to that in an FG cylinder. The material tailoring results presented here should help structural engineers and material scientists optimally design hollow cylinders and spheres comprised of inhomogeneous materials.     

ON THE USE OF FFT ALGORITHM FOR THE CIRCUMFERENTIAL CO-ORDINATE IN BOUNDARY ELEMENT FORMULATION OF AXISYMMETRIC PROBLEMS

A new numerical method is proposed for the boundary element analysis of axisymmetric bodies. The method is based on complex Fourier series expansion of boundary quantities in circumferential direction, which reduces the boundary element equation to an integral equation in (r–z) plane involving the Fourier coefficients of boundary quantities, where r and z are the co-ordinates of the (r, θ, z) cylindrical co-ordinate system. The kernels appearing in these integral equations can be computed effectively by discrete Fourier transform formulas together with the fast Fourier transform (FFT) algorithm, and the integral equations in (r–z) plane can be solved by Gaussian quadrature, which establishes the Fourier coefficients associated with boundary quantities. The Fourier transform solution can then be inverted into (r, θ, z) space by using again discrete Fourier transform formulas together with FFT algorithm. In the study, first we present the formulation of the proposed method which is outlined above. Then, the method is assessed by using three sample problems. A good agreement is observed in the comparisons of the predictions of the method with those available in the literature. It is further found that the proposed method provides considerable saving in computer time compared to existing methods of literature. © 1997 by John Wiley & Sons, Ltd.     

Effectiveness of mixing coaxial flows swirled in opposite directions

The mixing of coaxial turbulent flows swirled in opposite directions is experimentally studied. The effectiveness of this mixing is compared with mixing after an agitating grid.Notation z, r, cylindrical coordinate system - r₁, r₂ inside and outside radius of annular channel - H=r₂–r₁ radial gap in annular channel - y=(r–r₁)/H dimensionless radial coordinate - V(v_z, v_r, v) mean velocity vector - v, v_z pulsative components of velocity in the direction of the mean velocity vector and in the axial direction - P_*, P total and static pressure - , loss coefficients Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 407–413, September, 1981.