On modified displacement version of the non-linear theory of thin shells

We discuss the non-linear theory of thin shells expressed in terms of displacements of the shell reference surface as the only independent field variables. The formulation is based on the principle of virtual work postulated for the reference surface. In our approach: (1) the vector equilibrium equations are represented through components in the deformed contravariant surface base, and using the compatibility conditions the resulting tangential equilibrium equations are additionally simplified, (2) at the shell boundary the new scalar function of displacement derivatives is defined and new sets of four work-conjugate static and geometric boundary conditions are derived, as well as (3) for prescribed shell geometry all non-linear shell relations are generated automatically by two packages set up in Mathematica. The displacement boundary value problem and the associated homogeneous shell buckling problem are generated exactly without using any additional approximations following from errors of the constitutive equations. Both problems are extremely complex and available only in the computer memory. Such an approach allows us to account also for those a few supposedly small terms which may be critical for finding the correct buckling load of shells sensitive to imperfections. This approach is used in the accompanying paper by Opoka and Pietraszkiewicz [Opoka, S., Pietraszkiewicz, W., 2009. On refined analysis of bifurcation buckling for the axially compressed circular cylinder. International Journal of Solids and Structures, 46, 3111–3123.] to perform the refined numerical analysis of bifurcation buckling for the axially compressed circular cylinder.     

On one skew-symmetric problem of electroelasticity for inhomogeneous cylinder of finite length

In the present paper, static bending problem of the electroelasticity for an inhomogeneous cylinder of finite length with sliding fixed end-supports is investigated. The given boundary value problem is reduced to a system of 12 k (k = 1, 2, …) integro-differential equations. Expressions for the components characterizing the state of stress for the inhomogeneous cylinder are presented. Based on the developed analytical algorithm, extensive numerical investigations associated with the stress analysis of an inhomogeneous piezoceramic cylinder have been conducted. The results of these investigations are illustrated graphically, demonstrating the stress distributions in piezoceramic circular and elliptical cylinders with inclusions of various geometries.     

Thermal study of an array of inline horizontal cylinders below a nearly adiabatic ceiling

Steady state two-dimensional free convection heat transfer from a horizontal, isothermal cylinder in a horizontal array of cylinders consists of three isothermal cylinders, located underneath a nearly adiabatic ceiling is studied experimentally. A Mach–Zehnder interferometer is used to determine thermal field and smoke test is made to visualize flow field. Effects of the cylinders spacing to its diameter (S/D), and cylinder distance from ceiling to its diameter (L/D) on heat transfer from the centered cylinder are investigated for Rayleigh numbers from 1500 to 6000. Experiments are performed for an inline array configuration of horizontal cylinders of diameters D = 13 mm. Results indicate that due to the nearly adiabatic ceiling and neighboring cylinders, thermal plume resulted from the centered cylinder separates from cylinder surface even for high L/D values and forming recirculation regions. By decreasing the space ratio S/D, the recirculation flow strength increases. Also, by decreasing S/D, boundary layers of neighboring cylinders combine and form a developing flow between cylinders. The strength of developing flow depends on the cylinders Rayleigh number and S/D ratio. Due to the developing flow between cylinders, the vortex flow on the top of the centered cylinder appears for all L/D ratios and this vortex influences the value of local Nusselt number distribution around the cylinder.Variation of average Nusselt number of the centered cylinder depends highly on L/D and the trend with S/D depends on the value of Rayleigh number.     

Instantaneous flow field above the free end of finite-height cylinders and prisms

The flow above the free ends of surface-mounted finite-height circular cylinders and square prisms was studied experimentally using particle image velocimetry (PIV). Cylinders and prisms with aspect ratios of AR = 9, 7, 5, and 3 were tested at a Reynolds number of Re = 4.2 × 10⁴. The bodies were mounted normal to a ground plane and were partially immersed in a turbulent zero-pressure-gradient boundary layer, where the boundary layer thickness relative to the body width was δ/D = 1.6. PIV measurements were made above the free ends of the bodies in a vertical plane aligned with the flow centreline. The present PIV results provide insight into the effects of aspect ratio and body shape on the instantaneous flow field. The recirculation zone under the separated shear layer is larger for the square prism of AR = 3 compared to the more slender prism of AR = 9. Also, for a square prism with low aspect ratio (AR = 3), the influence of the reverse flow over the free end surface becomes more significant compared to that for a higher aspect ratio (AR = 9). For the circular cylinder, a cross-stream vortex forms within the recirculation zone. As the aspect ratio of the cylinder decreases, the reattachment point of the separated flow on the free end surface moves closer to the trailing edge. For both the square prism and circular cylinder cases, the instantaneous velocity vector field and associated in-plane vorticity field revealed small-scale structures mostly generated by the separated shear layer.     

URANS vs. experiments of flow induced motions of multiple circular cylinders with passive turbulence control

Two-dimensional Unsteady Reynolds-Average Navier–Stokes equations with the Spalart–Allmaras turbulence model are used to simulate the flow induced motions of multiple circular cylinders with passive turbulence control (PTC) in steady uniform flow. Four configurations with 1, 2, 3, and 4 cylinders in tandem are simulated and studied at a series of Reynolds numbers in the range of 30 000<Re<120 000. Simulation results are verified by experimental data measured in the Marine Renewable Energy Laboratory. Good agreement was observed between the values of vorticity, amplitude ratio, and frequency ratio predicted by numerical simulations and experimental measurements. The amplitude and frequency response show the initial and upper branches in vortex induced vibration (VIV), transition from VIV to galloping, and galloping branch for all PTC-cylinders. The maximum amplitude of 2.9 diameters for the first cylinder is achieved at Re=104 356 in the numerical results. Compared with the first cylinder, the VIV initial branch starts at higher Re for the downstream cylinders due to the presence of the upstream cylinder(s). 2P and 2P+2S vortex patterns are observed at Re=62 049 and Re=90 254 for the single PTC-cylinder. Furthermore, the shed vortices of the downstream cylinders are strongly disrupted and modified by the vortices shed from the upstream one in the cases of multiple PTC-cylinders.     

Influence of aspect ratio on the flow above the free end of a surface-mounted finite cylinder

The flow above the free end of a surface-mounted finite-height cylinder was studied in a low-speed wind tunnel using particle image velocimetry (PIV). Velocity measurements were made in vertical and horizontal measurement planes above the free end of finite cylinders of aspect ratios AR = 9, 7, 5 and 3, at a Reynolds number of Re = 4.2 × 10⁴. The relative thickness of the boundary layer on the ground plane was δ/D = 1.7. Flow separating from the leading edge formed a prominent recirculation zone on the free-end surface. The legs of the mean arch vortex contained within the recirculation zone terminate on the free-end surface on either side of the centreline. Separated flow from the leading edge attaches onto the upper surface of the cylinder along a prominent attachment line. Local separation downstream of the leading edge is also induced by the reverse flow and arch vortex circulation within the recirculation zone. As the cylinder aspect ratio is lowered from AR = 9 to AR = 3, the thickness of the recirculation zone increases, the arch vortex centre moves downstream and higher above the free-end surface, the attachment position moves downstream, and the termination points of the arch vortex move upstream. A lowering of the aspect ratio therefore results in accentuated curvature of the arch vortex line. Changes in aspect ratio also influence the vorticity generation in the near-wake region and the shape of the attachment line.     

Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test

This paper derives a new three-dimensional (3-D) analytical solution for the indirect tensile tests standardized by ISRM (International Society for Rock Mechanics) for testing rocks, and by ASTM (American Society for Testing and Materials) for testing concretes. The present solution for solid circular cylinders of finite length can be considered as a 3-D counterpart of the classical two dimensional (2-D) solutions by Hertz in 1883 and by Hondros in 1959. The contacts between the two steel diametral loading platens and the curved surfaces of a cylindrical specimen of length H and diameter D are modeled as circular-to-circular Hertz contact and straight-to-circular Hertz contact for ISRM and ASTM standards respectively. The equilibrium equations of the linear elastic circular cylinder of finite length are first uncoupled by using displacement functions, which are then expressed in infinite series of some combinations of Bessel functions, hyperbolic functions, and trigonometric functions. The applied tractions are expanded in Fourier–Bessel series and boundary conditions are used to yield a system of simultaneous equations. For typical rock cylinders of 54 mm diameter subjected to ISRM indirect tensile tests, the contact width is in the order of 2 mm (or a contact angle of 4°) whereas for typical asphalt cylinders of 101.6 mm diameter subjected to ASTM indirect tensile tests the contact width is about 10 mm (or a contact angle of 12°). For such contact conditions, 50 terms in both Fourier and Fourier–Bessel series expansions are found sufficient in yielding converged solutions. The maximum hoop stress is always observed within the central portion on a circular section close to the flat end surfaces. The difference in the maximum hoop stress between the 2-D Hondros solution and the present 3-D solution increases with the aspect ratio H/D as well as Poisson’s ratio ν. When contact friction is neglected, the effect of loading platen stiffness on tensile stress in cylinders is found negligible. For the aspect ratio of H/D = 0.5 recommended by ISRM and ASTM, the error in tensile strength may be up to 15% for both typical rocks and asphalts, whereas for longer cylinders with H/D up to 2 the error ranges from 15% for highly compressible materials, and to 60% for nearly incompressible materials. The difference in compressive radial stress between the 2-D Hertz solution or 2-D Hondros solution and the present 3-D solution also increases with Poisson’s ratio and aspect ratio H/D. In summary, the 2-D solution, in general, underestimates the maximum tensile stress and cannot predict the location of the maximum hoop stress which typically locates close to the end surfaces of the cylinder.     

Convergence and performance of the h- and p-extensions with mixed finite element C0-continuity formulations,for tension and buckling of a gradient elastic beam

Mixed formulations with C⁰-continuity basis functions are employed for the solution of some types of one-dimensional fourth- and sixth-order equations, resulting from axial tension and buckling of gradient elastic beams, respectively. A basic characteristic of gradient elasticity type equations is the appearance of boundary layers in the higher-order derivatives of the displacements (e.g., in the stress fields). This is due to the small parameters (related to the size of the microstructure) entering the governing equations. The proposed mixed formulations are based on generalizations of the well-known Ciarlet–Raviart mixed method, where the new main variables are related to second-order (or fourth order, for the buckling problem) derivatives of the displacement field. The continuous and discrete Babuška–Brezzi inf–sup conditions are established. The mixed formulations are numerically tested for both the uniform h- and p-extensions. With regard to the axial tension problem, the standard quasi-optimal rates of convergence are numerically verified in all cases (i.e., algebraic rate of convergence for the h-extension and exponential rate for the p-extension). On the other hand, the h-extension observed convergence rates of the critical (buckling) load for the second model problem are slightly higher than the theoretical ones found in the literature (especially for polynomial order p = 1). The respective observed rates of convergence of the buckling load for the p-extension are still exponential.     

Buckling behavior of Kelvin open-cell foams under [0 0 1], [0 1 1] and [1 1 1] compressive loads

This paper describes buckling modes and stresses of elastic Kelvin open-cell foams subjected to [0 0 1], [0 1 1] and [1 1 1] uniaxial compressions. Cubic unit cells and cell aggregates in model foams are analyzed using a homogenization theory of the updated Lagrangian type. The analysis is performed on the assumption that the struts in foams have a non-uniform distribution of cross-sectional areas as observed experimentally. The relative density is changed to range from 0.005 to 0.05. It is thus found that long wavelength buckling and macroscopic instability primarily occur under [0 0 1] and [0 1 1] compressions, with only short wavelength buckling under [1 1 1] compression. The primary buckling stresses under the three compressions are fairly close to one another and almost satisfy the Gibson–Ashby relation established to fit experiments. By also performing the analysis based on the uniformity of strut cross-sectional areas, it is shown that the non-uniformity of cross-sectional areas is an important factor for the buckling behavior of open-cell foams.     

Flow structure around perforated cylinders in shallow water

The experimental investigations were carried out in order to have detailed information on the flow structure around perforated cylinders using high-image density Particle Image Velocimetry technique in shallow water flow. The depth-averaged free-stream velocity was kept constant as U_∞=100 mm/s corresponding to the Reynolds number of Re=10 000 based on the perforated cylinder diameter. In order to analyze the effect of porosity, β on the flow structure, the porosities in the range of 0.1≤β≤0.8 with an increment of 0.1 were used and the results were compared with the bare cylinder case by means of velocity and vorticity contours, turbulent kinetic energy, Reynolds shear stress and streamline topologies. It was concluded that the porosity, β had a substantial effect on the control of large-scale vortical structures downstream of the cylinder in which the shear layers were elongated, fluctuations were significantly attenuated and formation of Karman Vortex Street was successfully prevented by the use of perforated cylinders.     

Separated flow structures around a cylindrical obstacle in a narrow channel

A detailed experimental study is performed on the separated flow structures around a low aspect-ratio circular cylinder (pin-fin) in a practical configuration of liquid cooling channel. Distinctive features of the present arrangement are the confinement of the cylinder at both ends, water flow at low Reynolds numbers (Re = 800, 1800, 2800), very high core flow turbulence and undeveloped boundary layers at the position of the obstacle. The horseshoe vortex system at the junctions between the cylinder and the confining walls and the near wake region behind the obstacle are deeply investigated by means of Particle Image Velocimetry (PIV). Upstream of the cylinder, the horseshoe vortex system turns out to be perturbed by vorticity bursts from the incoming boundary layers, leading to aperiodical vortex oscillations at Re = 800 or to break-away and secondary vorticity eruptions at the higher Reynolds numbers. The flow structures in the near wake show a complex three-dimensional behaviour associated with a peculiar mechanism of spanwise mass transport. High levels of free-stream turbulence trigger an early instabilization of the shear layers and strong Bloor–Gerrard vortices are observed even at Re = 800. Coalescence of these vortices and intense spanwise flow inhibit the alternate primary vortex shedding for time periods whose length and frequency increase as the Reynolds number is reduced. The inhibition of alternate vortex shedding for long time periods is finally related to the very large wake characteristic lengths and to the low velocity fluctuations observed especially at the lowest Reynolds number.     

Some exact solutions for rotating flows of a generalized Burgers’ fluid in cylindrical domains

The velocity field and the adequate shear stress corresponding to the flow of a generalized Burgers’ fluid model, between two infinite co-axial cylinders, are determined by means of Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a time dependent torsional shear to the fluid. The solutions that have been obtained, presented in series form in terms of usual Bessel functions J₁( ? ), J₂( ? ), Y₁( ? ) and Y₂( ? ), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for Burgers’, Oldroyd-B, Maxwell, second grade, Newtonian fluids and large-time transient solutions for generalized Burgers’ fluid are also obtained as special cases of the present general solutions. The effect of various parameters on large-time and transient solutions of generalized Burgers’ fluid is also discussed. Furthermore, for small values of the material parameters, λ₂ and λ₄ or λ₁, λ₂, λ₃ and λ₄, the general solutions corresponding to generalized Burgers’ fluids are going to those for Oldroyd-B and Newtonian fluids, respectively. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Free convection heat transfer from a horizontal fin attached cylinder between confined nearly adiabatic walls

Steady state two-dimensional free convection heat transfer from a horizontal, isothermal fin attached cylinder, located between nearly two adiabatic walls is studied experimentally using a Mach–Zehnder interferometer. Effects of the walls inclination angel (θ) on heat transfer from the cylinder is investigated for Rayleigh number ranging from 1000 to 15,500. Two cylinders with different diameters of D = 10 and 20 mm are used to cover wide Rayleigh range. Results indicate that, heat transfer phenomena differ for different Rayleigh number. For Rayleigh numbers lower than 5500, heat transfer rate from cylinder surface is lower than the heat transfer from a single cylinder. In this range by the use of walls, heat transfer from the cylinder decreases slightly and walls’ inclination does not change heat transfer rate from the cylinder surface. For Rayleigh number ranging from 5500 to 15,500, amount of heat transfer from the cylinder surface is less than that of a single cylinder. However, by adding nearly adiabatic walls to experimental model heat transfer mechanism differs and chimney effect between fin and walls increases the heat transfer rate from the cylinder surface. By increasing the walls inclination angel from 0° to 20°, the chimney effect between walls and fin diminishes and heat transfer rate from the cylinder surface is approaching to the heat transfer rate of fin attached cylinder without adiabatic walls.     

Effect of side ratio on fluid flow and heat transfer from rectangular cylinders using the PANS method

A computational study of heat transfer from rectangular cylinders is carried out. Rectangular cylinders are distinguished based on the ratio of the length of streamwise face to the height of the cross-stream face (side ratio, R). The simulations were performed to understand the heat transfer in a flow field comprising separation, reattachment, vortex shedding and stagnation. The Partially-Averaged Navier–Stokes (PANS) modeling approach is used to solve the turbulent flow physics associated and the wall resolve approach is used for the near wall treatment because of the flow separation involved. The simulations were performed using a finite volume based opensource software, OpenFOAM, at Reynolds number (Re) = 22,000 for rectangular cylinder at constant temperature kept in an air stream. Two critical side ratios were obtained, R = 0.62 and 3.0. At R = 0.62, the maximum value of the drag coefficient (C_d) = 2.681 was observed which gradually reduced by 54% at R = 4.0. The base pressure coefficient and global Nusselt number also attained the maximum value at R = 0.62 and from R = 2.5 to 3.0 a sharp discontinuous increase by 140% in the Strouhal number was observed. At R = 0.62, it was observed that the separated flow reattaches at the trailing edge after rolling over the side face and therefore increases the overall Nusselt number. The phase averaging was also performed to analyze the unsteady behavior of heat transfer.     

Analysis of non-axisymmetric wave propagation in a homogeneous piezoelectric solid circular cylinder of transversely isotropic material

A study concerning the propagation of free non-axisymmetric waves in a homogeneous piezoelectric cylinder of transversely isotropic material with axial polarization is carried out on the basis of the linear theory of elasticity and linear electro-mechanical coupling. The solution of the three dimensional equations of motion and quasi-electrostatic equation is given in terms of seven mechanical and three electric potentials. The characteristic equations are obtained by the application of the mechanical and two types of electric boundary conditions at the surface of the piezoelectric cylinder. A novel method of displaying dispersion curves is described in the paper and the resulting dispersion curves are presented for propagating and evanescent waves for PZT-4 and PZT-7A piezoelectric ceramics for circumferential wave numbers m = 1, 2, and 3. It is observed that the dispersion curves are sensitive to the type of the imposed boundary conditions as well as to the measure of the electro-mechanical coupling of the material.     

Parametric instabilities of the radial motions of non-linearly viscoelastic shells under pulsating pressures

This paper treats the classical problem of radial motions of cylindrical and spherical shells under pulsating pressures. The novelty in this work is that the shells are taken to be non-linearly viscoelastic (of strain-rate type). It is remarkable that this classical problem, which does not treat the loss of stability to non-radial motions (but which is essential for such treatments), has such a rich dynamics due to the often neglected effects of non-linear material response, to the role of prestress under the action of the mean pressure, and to the different effects of pressure on cylindrical and spherical shells. The study of radial motions near primary resonance (when the frequency of the pulsating pressure is near the natural frequency about an equilibrium state under a constant pressure) gives formulas ensuring that the motions are of hardening or softening type depending on the constitutive functions and whether the constant mean pressure is compressive or inflational. The method of multiple scales gives asymptotic formulas for the principal parametric instability regions (Mathieu tongues) and for the stable and unstable motions at twice the forcing frequency, which closely agree with those obtained by numerical continuation methods. The dependence of frequency on amplitude and the form of instability regions are critically influenced by deviations (even very slight deviations) of material response from that of linearly viscoelastic shells, by the constant mean pressure, and by the type of shell. This paper exhibits the rich diversity of postcritical periodic motions.     

Cracked infinite cylinder with two rigid inclusions under axisymmetric tension

This paper considers the problem of an axisymmetric infinite cylinder with a ring shaped crack at z = 0 and two ring-shaped rigid inclusions with negligible thickness at z = ±L. The cylinder is under the action of uniformly distributed axial tension applied at infinity and its lateral surface is free of traction. It is assumed that the material of the cylinder is linearly elastic and isotropic. Crack surfaces are free and the constant displacements are continuous along the rigid inclusions while the stresses have jumps. Formulation of the mixed boundary value problem under consideration is reduced to three singular integral equations in terms of the derivative of the crack surface displacement and the stress jumps on the rigid inclusions. These equations, together with the single-valuedness condition for the displacements around the crack and the equilibrium equations along the inclusions, are converted to a system of linear algebraic equations, which is solved numerically. Stress intensity factors are calculated and presented in graphical form.     

Circumferential temperature distribution during nucleate pool boiling outside smooth and modified horizontal tubes

In the work an approach to avoid a circumferential temperature distribution existing during nucleate pool boiling on a horizontal cylinder within low heat flux densities is presented. The idea of the approach is local heat transfer enhancement by a porous layer application on a part of the heating surface. An experiment on nucleate pool boiling heat transfer from horizontal cylinders to saturated R141b and water under atmospheric pressure is reported. Experiments have been conducted using stainless steel tubes with the outside diameter between 8 mm and 23 mm with the active length of 250 mm. The outside surface of the tubes was smooth or partially coated with a porous metallic layer. In particular, measurements of inside circumferential temperature distribution have been performed.     

Theoretical stress analysis of intersecting cylindrical shells subjected to external loads transmitted through branch pipes

A thin shell theoretical solution of two normally intersecting cylindrical shells subjected to thrust-out force and three kinds of moments transmitted through branch pipes is presented in this paper. The solutions of modified Morley equation, which can be applicable up to ρ₀ = d/D ⩽ 0.8 and λ = d/(DT)^1/2 ⩽ 8 and the order of accuracy is raised to O(T/D), for the four loading cases are given. The accurate continuity conditions of generalized forces and displacements at the intersecting curve of two cylindrical shells for the four loading cases and the condition of the uniqueness of displacements are derived in this paper. The presented results are verified by experimental and numerical results successfully. They are in agreement with WRC Bulletin 297 when d/D is small.     

Coherent and turbulent processes in the bistable regime around a tandem of cylinders including reattached flow dynamics by means of high-speed PIV

The turbulent flow around two cylinders in tandem at the sub-critical Reynolds number range of order of 10⁵ and pitch to diameter ratio of 3.7 is investigated by using time-resolved Particle Image Velocimetry (TRPIV) of 1 kHz and 8 kHz. The bi-stable flow regimes including a flow pattern I with a strong vortex shedding past the upstream and the downstream cylinder, as well as a flow pattern II corresponding to a weak alternating vortex shedding with reattachment past the upstream cylinder are investigated. The structure of this “reattachment regime” has been analyzed in association with the vortex dynamics past the downstream cylinder, by means of POD and phase-average decomposition. These elements allowed interconnection among all the measured PIV planes and hence analysis of the reattachment structure and the flow dynamics past both cylinders. The results highlight fundamental differences of the flow structure and dynamics around each cylinder and provide the ‘gap’ flow nature between the cylinders. Thanks to a high-speed camera of 8 kHz, the shear-layer vortices tracking has been possible downstream of the separation point and the quantification of their shedding frequency at the present high Reynolds number range has been achieved. This issue is important regarding fluid instabilities involved in the fluid–structure interaction of cylinder arrays in nuclear reactor systems, as well as acoustic noise generated from the tandem cylinders of a landing gear in aeronautics.