On Extension and Torsion of Strain-Stiffening Rubber-Like Elastic Circular Cylinders

This paper is concerned with investigation of the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic nonlinearly elastic materials. Our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two particular phenomenological constitutive models for such materials that reflect limiting chain extensibility at the molecular level. The axial stretch γ and twist that can be sustained in cylinders composed of such materials are shown to be constrained in a coupled fashion. It is shown that, in the absence of an additional axial force, a transition value γ=γ _t of the axial stretch exists such that for γ<γ _t , the stretched cylinder tends to elongate on twisting whereas for γ>γ _t , the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models such as the Mooney-Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. We also obtain results for materials modeled by the well-known exponential strain-energy widely used in biomechanics applications. This model reflects a strain-stiffening that is less abrupt than that for the limiting chain extensibility models. Surprisingly, it turns out that the results in this case are somewhat more complicated. For a fixed stiffening parameter, provided that the stretch is sufficiently small, the stretched bar always tends to elongate on twisting in the absence of an additional axial force. However, for sufficiently large stretch, the cylinder tends to shorten on undergoing sufficiently small twist but then tends to elongate on further twisting. These results are of interest in view of the widespread use of exponential models in the context of the mechanics of soft biological tissues. The special case of pure torsion is also briefly considered. In this case, the resultant axial force required to maintain pure torsion is compressive for all the models discussed here. In the absence of such a force, the bar would elongate on twisting reflecting the celebrated Poynting effect.      

Numerical simulation of flows around two circular cylinders by mesh‐free least square‐based finite difference methods

In this paper, the mesh‐free least square‐based finite difference (MLSFD) method is applied to numerically study the flow field around two circular cylinders arranged in side‐by‐side and tandem configurations. For each configuration, various geometrical arrangements are considered, in order to reveal the different flow regimes characterized by the gap between the two cylinders. In this work, the flow simulations are carried out in the low Reynolds number range, that is, Re=100 and 200. Instantaneous vorticity contours and streamlines around the two cylinders are used as the visualization aids. Some flow parameters such as Strouhal number, drag and lift coefficients calculated from the solution are provided and quantitatively compared with those provided by other researchers. Copyright © 2006 John Wiley & Sons, Ltd.     

Flow between parallel walls containing the lines of neutrally buoyant circular cylinders

The motions of a single and two lines of neutrally buoyant circular cylinders in fluid between flat parallel walls are numerically investigated over the range of the Reynolds number of 12 < Re < 96, the ratio of the diameter of the cylinder D_s to the channel width D of 0.25≤D_s/D≤0.5, and the ratio of the streamwise spacing of the cylinders L to the channel width of 0.75≤L/D≤2. The lattice Boltzmann method is used for computations of the fluid phase and the cylinders are moved according to Newton’s law of motion. The Segré–Silberberg effect is found for both a single and two lines of cylinders. It is also found that for two lines of cylinders with D_s/D=0.25 and L/D=1, the equilibrium positions of the two lines are arranged to be staggered with respect to each other in the flow direction. The effects of the Reynolds number Re, D_s/D, and L/D on the equilibrium position of the lines of cylinders and on the friction factor of the cylinder–fluid mixture are presented and discussed.     

Flow-induced oscillation of two circular cylinders in tandem arrangement at low Re

Results are presented for flow-induced vibrations of a pair of equal-sized circular cylinders of low nondimensional mass (m^*=10) in a tandem arrangement. The cylinders are free to oscillate both in streamwise and transverse directions. The Reynolds number, based on the free-stream speed and the diameter of the cylinders, D is 100 and the centre-to-centre distance between the cylinders is 5.5D. The computations are carried out for reduced velocities in the range 2≤U^*≤15. The structural damping is set to zero for enabling maximum amplitudes of oscillation. A stabilized finite element method is utilized to carry out the computations in two dimensions. Even though the response of the upstream cylinder is found to be qualitatively similar to that of an isolated cylinder, the presence of a downstream cylinder is found to have significant effect on the behaviour of the upstream cylinder. The downstream cylinder undergoes very large amplitude of oscillations in both transverse and streamwise directions. The maximum amplitude of transverse response of the downstream cylinder is quite similar to that of a single cylinder at higher Re beyond the laminar regime. Lock-in and hysteresis are observed for both upstream and downstream cylinders. The downstream cylinder undergoes large amplitude oscillations even beyond the lock-in state. The phase between transverse oscillations and lift force suffers a 180 jump for both the cylinders almost in the middle of the synchronization regime. The phase between the transverse response of the two cylinders is also studied. Complex flow patterns are observed in the wake of the freely vibrating cylinders. Based on the phase difference and the flow patterns, the entire flow range is divided into five sub-regions.     

Numerical Fluid Flow Analysis for Aerodynamic Response Characteristics of Tandem Circular Cylinders

The aerodynamic characteristics of tandem cables of cable-stayed bridges have become an increasingly serious problem with increments in span length. In order to reduce the construction cost and maintenance of cables, tandem cables have been adopted for cable-stayed bridges. These cables, however, have aerodynamic response characteristics such as wake-galloping. Therefore, a method to suppress wake-galloping in tandem cables is required. The purpose of this study is to investigate the characteristics of the wake-galloping phenomenon of tandem cables of cable-stayed bridges using numerical fluid flow analysis. The flow around the oscillating tandem circular cylinders modeled on tandem cables is calculated. The flow field is treated as an incompressible viscous flow. The Arbitrary Lagrangian-Eulerian (ALE) method is employed to solve the flow field around the cylinders, and the three-step Taylor-Galerkin method, which is based on a fractional step finite element method, is adopted for discretization.     

Force coefficients and Strouhal numbers of three circular cylinders subjected to a cross-flow

In this paper, wind tunnel experiments were conducted to measure the mean force coefficients and Strouhal numbers for three circular cylinders of equal diameters in an equilateral-triangular arrangement when subjected to a cross-flow. These experiments were carried out at five subcritical Reynolds numbers ranging from 1.26 × 10⁴ to 6.08 × 10⁴. The pressure distributions on the surface of the cylinders were measured using pressure transducers. Furthermore, the hot-wire anemometer was employed to measure the vortex shedding frequencies behind each cylinder. Six spacing ratios (l/d) varying from 1.5 to 4 were investigated. It is observed that for l/d > 2, the upstream cylinder experiences a lower mean drag coefficient compared with the downstream cylinders. The minimum values of the drag coefficient for the downstream cylinders occur at l/d = 1.5 and l/d = 2, because there is no vortex shedding from the foregoing cylinders. Also, the value of the pressure coefficient behind the upstream cylinder reduces by increasing l/d. Moreover, by decreasing the value of l/d, the Strouhal number for the upstream cylinder increases. It can be concluded that the flow pattern and aerodynamic coefficients are basically dependent on l/d; in other words, decreasing l/d results in an increase in the effects of the flow interference between the cylinders.     

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 + , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.     

Numerical Simulation of Aerodynamic Characteristics of Two Rectangular Cylinders in Side-by-side Arrangement

We present numerical results of flows around two rectangular cylinders in side-by-side arrangement by three-dimensional computations. The two rectangular cylinders are arranged with various distances between the cylinders. The three-dimensional flow structures around two rectangular cylinders denote significant features depending on the distance between the cylinders. In our computations, the Reynolds number Re is set as 10,000, and both the aspect ratio of section of two rectangular cylinders and the distance between the cylinders are considered as parameters to calculate the flow around two rectangular cylinders. The obtained numerical results are also compared with experimental data.     

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem has attracted attention in investigations of the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. Here we investigate similar issues for fiber-reinforced transversely-isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The classic Poynting effect found for rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely-isotropic materials considered here. For sufficiently large anisotropy and under certain conditions on the amount of twist, a reverse-Poynting effect is demonstrated where the cylinder tends to shorten on twisting The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft tissues.     

Self-sustained oscillations between two tandem cylinders at Reynolds number 1,000

This study focuses on the self-sustained oscillatory flow characteristics between two tandem circular cylinders of equal diameter placed in a uniform inflow. The Reynolds number (Re _D ), based on the cylinder diameter, was around 1,000 and all experiments were performed in a recirculating water channel. The streamwise distance between two tandem cylinders ranged within 1.5 ≤ X _c/D ≤ 7.0. Here X _c denotes the center-to-center distance between two tandem cylinders. For all experiments studied herein, quantitative velocity measurements were performed using hot-film anemometer and the LDV system. The laser sheet technique was employed for qualitative flow visualization. The wavelet transform was applied to elucidate the temporal variation and phase difference between two spectral components of the velocity signals detected in the flow field. The remarkable finding was that when two tandem circular cylinders were spaced at a distance within 4.5 ≤ X _c/D ≤ 5.5, two symmetrical unstable shear layers with a certain wavelength were observed to impinge onto the downstream cylinder. The responding frequency (f _u ), measured between these two cylinders, was much higher than the natural shedding frequency behind a single isolated cylinder at the same Re _D . This responding frequency decreased as the distance X _c/D increased. Not until X _c/D ≥ 6.0, did it recover to the natural shedding frequency behind a single isolated cylinder. Between two tandem cylinders, the Strouhal numbers (St _c = f _u X _c/U_c) maintained a nearly constant value of 3, indicating the self-sustained oscillating flow characteristics with a wavelength X _c/3. Here U _c is the convection speed of the unstable shear layers between two tandem cylinders. At Re _D = 1,000, the self-sustained oscillating characteristics between two tandem circular cylinders were proven to exhibit a sustained flow pattern, not just a sporadic phenomenon.     

ENDOCHRONIC ANALYSIS FOR COMPRESSIVE BUCKLING OF THIN－WALLED CYLINDERS IN YIELD REGION

ＥＮＤＯＣＨＲＯＮＩＣＡＮＡＬＹＳＩＳＦＯＲＣＯＭＰＲＥＳＳＩＶＥＢＵＣＫＬＩＮＧＯＦＴＨＩＮ－ＷＡＬＬＥＤＣＹＬＩＮＤＥＲＳＩＮＹＩＥＬＤＲＥＧＩＯＮＰｅｎｇＸｉａｎｇｈｅ（彭向和）ＣｈｅｎＹｕａｎｑｉａｎｇ（陈元强）ＺｅｎｇＸｉａｎｇｇｕｏ（曾祥...     

A numerical investigation of the flow over a pair of identical square cylinders in a tandem arrangement

This paper describes a numerical study of the two‐dimensional and three‐dimensional unsteady flow over two square cylinders arranged in an in‐line configuration for Reynolds numbers from 40 to 1000 and a gap spacing of 4D, where D is the cross‐sectional dimension of the cylinders. The effect of the cylinder spacing, in the range G = 0.3D to 12D, was also studied for selected Reynolds numbers, that is, Re = 130, 150 and 500. An incompressible finite volume code with a collocated grid arrangement was employed to carry out the flow simulations. Instantaneous and time‐averaged and spanwise‐averaged vorticity, pressure, and streamlines are computed and compared for different Reynolds numbers and gap spacings. The time averaged global quantities such as the Strouhal number, the mean and the RMS values of the drag force, the base suction pressure, the lift force and the pressure coefficient are also calculated and compared with the results of a single cylinder. Three major regimes are distinguished according to the normalized gap spacing between cylinders, that is, the single slender‐body regime (G < 0.5), the reattach regime (G < 4) and co‐shedding or binary vortex regime (G ≥4). Hysteresis with different vortex patterns is observed in a certain range of the gap spacings and also for the onset of the vortex shedding. Copyright © 2011 John Wiley & Sons, Ltd.     

层流条件下并列三圆柱涡激振动响应与尾流形态

应用基于嵌入式压强-力迭代的高精度浸入边界法研究等间距并列三圆柱涡激振动。其中,雷诺数Re=100,间距比T/D=2.0～5.0,圆柱质量比m*=2.0,折合流速Ur=2.0～10.0,忽略振动系统的阻尼且三圆柱仅横向振动。研究发现,圆柱的振动响应随折合流速的增加呈现初始响应分支和下端响应分支两种模式;振幅响应出现不连续现象,且随着间距比的增加,该不连续现象对应的折合流速增加;尾流模式与间距比和折合流速密切相关。共发现六种尾流形态,分别为窄宽窄尾流、不规律尾流、反相同步尾流、调制尾流、同相同步尾流和偏斜尾流。总结并绘制了尾流形态在参数空间[Ur,T/D]内的分区图。     

Experimental Investigation of the Transverse Self-Oscillations of Circular Cylinders Mounted with a Narrow Clearance in a Plane Channel

The results of an investigation of the flow past and the behavior of free bluff bodies mounted in pipes and channels with a narrow clearance, conducted in the Institute of Mechanics of Moscow State University, are presented. The drag of circular cylinders of different size and mass in a circulation-free water flow in a plane channel of rectangular cross-section was studied in the transverse self-oscillation regime. The experiments were conducted for Reynolds numbers based on the cylinder diameter 1.7 ? 10⁴ ≤ Re ≤ 7.2 ? 10⁴, relative clearances \(\bar S\) based on a cross-sectional area ranging from 0.76 to 0.9, and cylinder-to-water density ratios ρ _c /ρ ranging from 1.29 to 8.2. Only the case of intense transverse self-oscillations accompanied by impact interaction with the channel wall was considered. The dependence of the period-average cylinder drag coefficient C _x on the basic dimensionless relevant parameters is obtained. The dependence of the dimensionless self-oscillation frequency determined in [1] is refined. The kinematic and dynamic features of the flows past spheres in cylindrical pipes and cylinders in plane channels are compared in the transverse self-oscillation regime.     

Laminar forced convection heat transfer from isothermal cylinders with active ends and different aspect ratios in axial air flows

In this article a semi-analytical approach is employed to obtain dimensionless heat transfer correlations for forced convection from isothermal circular cylinders with active ends and different aspect ratios (l/d ￡ 8) (l/d \le 8) in laminar axial air flows. Then, using the present results and previous works, the modeling is extended to higher aspect ratios (l/d 3 8) (l/d \ge 8) ) as long as the entire flow field remains completely laminar. Validations of the present work are done not only with the available data on drag coefficients but with previous works for long cylinders with inactive ends and long spheroids. Two general correlations are also developed for a rough estimate of forced convection heat transfer from isothermal cylinders with active ends and arbitrary aspect ratios in the range of \frac12 ￡ \fracld ￡ 8 \frac{1}{2} \le \frac{l}{d} \le 8 and l/d 3 8 l/d \ge 8 .     

TANDEM CYLINDERS IN IMPULSIVELY STARTED FLOW

The impulsively started flow field for circular cylinders of equal diameter arranged in tandem was investigated using flow visualization and particle image velocimetry (PIV), over a longitudinal pitch ratio range ofL /D=1·0–3·0, and for Reynolds numbers from Re=1200–3800. The PIV technique was used to obtain a time history of the instantaneous in-plane vorticity field from the moment of impulsive start, from which the spatial and temporal development of the flow was studied. Measurements of vortex strength and vortex position relative to the cylinders were obtained from these data. Three types of fluid behaviour were identified based on L/D: single bluff-body behaviour when the cylinders are in contact, constrained streamwise growth and lateral expansion of the gap recirculation zones at small and intermediate L/D, and independent formation of recirculation zones similar to a single impulsively started circular cylinder at larger L/D.     

Drag measurements on long thin cylinders at small angles and high Reynolds numbers

Measurements of the drag caused by turbulent boundary layer mean wall shear stress on cylinders at small angles of attack and high length Reynolds numbers (8×10⁶<Re_L<6×10⁷) are presented. The use of a full-scale, high-speed towing tank enabled the development of turbulent boundary layers on cylinders made of stainless steel, aluminum, titanium, and polyvinyl chloride. The diameter of all cylinders in this experiment was 12.7 mm; two cylinder lengths, 3.05 m and 6.10 m, were used, corresponding to aspect ratio values L/a=480 and 960, respectively. Materials of various densities were towed at critical angles, resulting in linear cylinder geometry for tow speeds ranging from 2.6 m/s to 20.7 m/s and angles between 0° and 12°. Towing angles were measured with digital photography, and streamwise drag was measured with a strut-mounted load cell at the tow point. The measured tangential drag was very sensitive to small increases in angle at all tow speeds. A momentum thickness length scale is proposed to scale the tangential drag coefficient. The effects of the cross-flow resulting from the small angles of tow have a significant effect on the tangential drag coefficient values. A scaling for the orthogonal force on the cylinders was determined and provides a correction to published normal drag coefficient values for pure cross-flow. The presence of the axial turbulent boundary layer has a significant effect on these orthogonal forces.     

Passive control of wake flow by two small control cylinders at Reynolds number 80

Passive control of the wake behind a circular cylinder in uniform flow is studied by numerical simulation at Re_D=80. Two small control cylinders are placed symmetrically along the separating shear layers at various stream locations. In the present study, the detailed flow mechanisms that lead to a significant reduction in the fluctuating lift but maintain the shedding vortex street are clearly revealed. When the stream locations lie within 0.8≤X_C/D≤3.0, the alternate shedding vortex street remains behind the control cylinders. In this case, the symmetric standing eddies immediately behind the main cylinder and the downstream delay of the shedding vortex street are the two primary mechanisms that lead to a 70–80% reduction of the fluctuating lift on the main cylinder. Furthermore, the total drag of all the cylinders still has a maximum 5% reduction. This benefit is primarily attributed to the significant reduction of the pressure drag on the main cylinder. Within X_C/D>3.0, the symmetry of the standing eddy breaks down and the staggered vortex street is similar to that behind a single cylinder at the same Reynolds number. In the latter case, the mean pressure drag and the fluctuating lift coefficients on the main cylinder will recover to the values of a single cylinder.     

Role of the shear layer instability in the near wake behavior of two side-by-side circular cylinders

Wakes, and their interaction behind two parallel cylinders lying in a plane perpendicular to the flow, have been investigated experimentally in the sub-critical Reynolds number regime. The experiments were performed in a water channel using laser Doppler velocimetry. The gap between the two cylinders was less than the cylinder diameter, a geometry referred to as strong interaction configuration. In this case the blockage is strong and a gap-jet appears between the cylinders. Two flow regimes of the near wake region have been identified: one below a critical Reynolds number Re _c ]1000;1700[, where the gap jet is stably deflected to one side and the double near-wake becomes asymmetric; the other, above Re _c, where the gap-jet deflection is unstable and a random flopping phenomenon takes place. When Re<Re _c, two different Strouhal numbers are identified, related to the Kármán vortex shedding behind each cylinder. When Re>Re _c, a third frequency appears in the near wake, related to the development of Kelvin-Helmholtz vortices in the separated shear layer of the cylinders [Prasad A, Williamson CHK (1997) J Fluid Mech 333:375]. The observed flopping behavior is attributed to the birth of these Kelvin-Helmholtz instabilities and their intermittent nature. Further downstream, beyond about five cylinder diameters, the random flopping flow phenomena disappear while a slightly asymmetric single wake persists. It is characterized by a Strouhal number St=0.13, a value that one would normally measure behind a single cylinder of twice its diameter.