考虑范德华力曲率效应的双壁碳纳米管外压屈曲

针对双壁碳纳米管外压屈曲问题,研究了层间范德华力的曲率效应对临界外压的影响。应用弹性双层圆柱壳模型,考虑层间范德华力不仅与层间距有关而且与挠度曲率的变化有关,导出了外压屈曲临界压力解析公式。计算得出在不同半径、不同长细比下,外压屈曲临界压力的数值结果,并与经典壳的结果和忽略范德华力曲率效应的结果做了比较。结果显示,对于小半径的双壁碳纳米管曲率效应对外压屈曲有效明显的影响。     

Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells

This paper models van der Waals (vdW) force for axially compressed multi-walled carbon nanotubes (CNTs), whereby each tube is treated as a cylindrical shell continuum. Explicit formulas are derived for predicting the critical axial strain of a triple-walled CNT using a more refined vdW model. The analysis of a cylindrical shell continuum model of multi-walled CNTs using this refined vdW force model is carried out to study the influence of the effect of vdW interaction between different layers of a CNT and the size effect of a CNT on the vdW interaction. It is shown herein that the greatest contribution to the vdW interaction comes from the adjacent layers and the contribution from a remote layer may be neglected. The vdW interaction is found to be strongly dependent on the radius of the tube, especially when the radius is small enough (<7 nm). When the radius is large enough (>40 nm), the vdW interaction coefficient c_ij can be taken as a constant value (i.e. independent of radius). However, these constant values are different for the vdW interaction between two different layers of a multi-walled CNT. The effect of the vdW interaction on the critical axial strain of a triple-walled CNT for the cases of before and after buckling is also examined for various innermost radii.     

The thermal effect on axially compressed buckling of a double-walled carbon nanotube

The thermal effect on axially compressed buckling of a double-walled carbon nanotube is studied in this paper. The effects of temperature change, surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model with thermal effect is presented for axially compressed buckling of a double-walled carbon nanotube embedded in an elastic matrix under thermal environment. Based on the model, an explicit formula for the critical axial stress is derived in terms of the buckling modes of the shell and the parameters that indicate the effects of temperature change, surrounding elastic medium and the van der Waals forces. Based on that, some simplified analysis is carried out to estimate the critical axial stress for axially compressed buckling of the double-walled carbon nanotube. Numerical results for the general case are obtained for the thermal effect on axially compressed buckling of a double-walled carbon nanotube. It is shown that the axial buckling load of double-walled carbon nanotube under thermal loads is dependent on the wave number of axially buckling modes. And a conclusion is drawn that at low and room temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube increase as the value of temperature change increases, while at high temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube decrease as the value of temperature change increases.     

Laminar,transitional and turbulent annular flow of drag-reducing polymer solutions

Mean and rms axial velocity-profile data obtained using laser Doppler anemometry are presented together with pressure-drop data for the flow through a concentric annulus (radius ratio κ = 0.506) of a Newtonian (a glycerine–water mixture) and non-Newtonian fluids—a semi-rigid shear-thinning polymer (a xanthan gum) and a polymer known to exhibit a yield stress (carbopol). A wider range of Reynolds numbers for the transitional flow regime is observed for the more shear-thinning fluids. In marked contrast to the Newtonian fluid, the higher shear stress on the inner wall compared to the outer wall does not lead to earlier transition for the non-Newtonian fluids where more turbulent activity is observed in the outer wall region. The mean axial velocity profiles show a slight shift (～5%) of the location of the maximum velocity towards the outer pipe wall within the transitional regime only for the Newtonian fluid.     

DYNAMIC BUCKLING OF DOUBLE-WALLED CARBON NANOTUBES UNDER STEP AXIAL LOAD

An approximate method is presented in this paper for studying the dynamic buckling of double-walled carbon nanotubes （DWNTs） under step axial load. The analysis is based on the continuum mechanics model, which takes into account the van der Waals interaction between the outer and inner nanotubes. A buckling condition is derived, from which the critical buckling load and associated buckling mode can be determined. As examples, numerical results are worked out for DWNTs under fixed boundary conditions. It is shown that, due to the effect of van der Waals forces, the critical buckling load of a DWNT is enhanced when inserting an inner tube into a single-walled one. The paper indicates that the critical buckling load of DWNTs for dynamic buckling is higher than that for static buckling. The effect of the radii is also examined. In addition, some of the results are compared with the previous ones.     

Buckling pressure of “pumpkin” balloons

This paper presents a computational study of the critical buckling pressure of pumpkin balloons, which consist of a thin, compliant membrane constrained by stiff meridional tendons. The n-fold symmetric shape of a pumpkin balloon with n identical lobes is exploited by adopting a symmetry-adapted coordinate system, which leads to the tangent stiffness matrix in an efficient block-diagonal form; the smallest eigenvalue of a particular block leads to the buckling pressure for the balloon. Two different types of balloon design are considered. Extensive results are obtained for the buckling pressures of a set of 10 m diameter experimental balloons and also for an 80 m diameter flight balloon. The key findings are as follows: the same type of buckling mode, forming four circumferential waves is critical for most of the balloons that have been analysed; balloons with flatter lobes are more stable, and the buckling pressure varies with an inverse power-law of the number of lobes; increasing the Young’s modulus, the Poisson’s ratio of the membrane, or the diameter of the end fitting has the effect of increasing the buckling pressure; but increasing the axial stiffness of the tendons has the effect of decreasing the buckling pressure.     

Experimental study on the compressive behaviors of CFRP tube-encased concrete columns

Nine square concrete columns including 6 CFRP/ECCs and 3 concrete columns are prepared, which have cross-section of 200 mm × 200 mm and height of 600 mm. The CFRP tubes with fibers oriented at hoop direction were manufactured to have 3 or 5 layers of CFRP with 10 mm, 20 mm, or 40 mm rounding corner radii at vertical edges. A 100 mm overlap in the direction of fibers was provided to ensure proper bond. Uniaxial compression tests were conducted to investigate the compressive behavior. It is evident that the CFRP tube confinement can improve the behavior of concrete core, in terms of axial compressive strength or axial deformability. Test results show that the stress-strain behavior of CFRP/ECCs vary with different confinement parameters, such as the number of confinement layers and the rounding corner radius.     

On refined analysis of bifurcation buckling for the axially compressed circular cylinder

We present extensive numerical results of bifurcation buckling analysis of the axially compressed circular cylinder. The analysis is based on the modified displacement version of the non-linear theory of thin elastic shells developed by Opoka and Pietraszkiewicz [Opoka, S., Pietraszkiewicz, W., 2009. On modified displacement version of the non-linear theory of thin shells. International Journal of Solids and Structures, 46, 3103–3110.]. To solve the buckling problem we apply the separation of variables and expansion of all fields into Fourier series in circumferential direction, with subsequent accurate calculations of eigenvalues of determinants of corresponding 8 × 8 complicated matrices. The numerical analysis of the buckling load is performed for the cylinders with length-to-diameter ratio in the range (0.05, 60), with eight sets of incremental work-conjugate boundary conditions analogous to those used in the literature and partly summarized in the book by Yamaki [Yamaki, N., 1984. Elastic Stability of Circular Cylindrical Shells. Elsevier, Amsterdam], and additionally with six sets of boundary conditions not discussed in the literature yet. The results allow us to formulate several important conclusions, such as: (a) omission in the non-linear BVP small terms of the order of error introduced by the error of constitutive equations leads to overestimated buckling loads for long cylinders with clamped boundaries; (b) for some relaxed boundary conditions the buckling load decreases for short cylinders with decrease of the cylinder length; (c) the results for additional six sets of boundary conditions reveal existence of several new cases, in which by relaxing geometric boundary conditions the buckling load falls down to about one half of the classical value in a wide range of the cylinder length-to-diameter ratios.     

Evolution of turbulence characteristics from straight to curved pipes

Fully developed, statistically steady turbulent flow in straight and curved pipes at moderate Reynolds numbers is studied in detail using direct numerical simulations (DNS) based on a spectral element discretisation. After the validation of data and setup against existing DNS results, a comparative study of turbulent characteristics at different bulk Reynolds numbers Re_b = 5300 and 11,700, and various curvature parameters κ = 0, 0.01, 0.1 is presented. In particular, complete Reynolds-stress budgets are reported for the first time. Instantaneous visualisations reveal partial relaminarisation along the inner surface of the curved pipe at the highest curvature, whereas developed turbulence is always maintained at the outer side. The mean flow shows asymmetry in the axial velocity profile and distinct Dean vortices as secondary motions. For strong curvature a distinct bulge appears close to the pipe centre, which has previously been observed in laminar and transitional curved pipes at lower Re_b only. On the other hand, mild curvature allows the interesting observation of a friction factor which is lower than in a straight pipe for the same flow rate.All statistical data, including mean profile, fluctuations and the Reynolds-stress budgets, is available for development and validation of turbulence models in curved geometries.     

Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction

Explicit formulas are derived for the van der Waals (vdW) interaction between any two layers of a multi-walled carbon nanotube (CNT). Based on the derived formulas, an efficient algorithm is established for the buckling analysis of multi-walled CNTs, in which individual tubes are modeled as a continuum cylindrical shell. The explicit expressions are also derived for the buckling of double-walled CNTs. In previous studies by Ru (J. Appl. Phys. 87 (2000b) 7227) and Wang et al. (Int. J. Solids Struct. 40 (2003) 3893), only the vdW interaction between adjacent two layers was considered and the vdW interaction between the other two layers was neglected. The vdW interaction coefficient was treated as a constant that was not dependent on the radii of the tubes. However, the formulas derived herein reveal that the vdW interaction coefficients are dependent on the change of interlayer spacing and the radii of the tubes. With the increase of radii, the coefficients approach constants, and the constants between two adjacent layers are about 10% higher than those reported by Wang et al. (Int. J. Solids. Struct. 40 (2003) 3893). In addition, the numerical results show that the vdW interaction will lead to a higher critical buckling load in multi-walled CNTs. The effect of the tube radius on the critical buckling load of a multi-walled CNT is also examined.     

Measurements of mean and turbulence quantities in the curved wake of an airfoil

Measurements of mean and turbulence quantities are presented for a curved wake of an airfoil. The wake is generated by placing a NACA 0012 airfoil of 0.150 m chord length at one chord length upstream of a 90° bend. The bend has a square cross-section of 0.457 m × 0.457 m, a mean radius-to-height ratio of R/H=1.17, and concave and convex radii of curvature 0.764 and 0.307 m, respectively. In addition to streamwise curvature, the wake is subjected to varying streamwise and radial pressure. The measurements were carried out at mainstream air velocities of 10, 15 and 20 m/s. The results are presented for the mean streamwise velocity, five components of turbulence stresses, the calculated wake half-width and the maximum velocity defect. The results showed the formation of an asymmetric wake about the wake centreline, with a larger wake half-width on the inner side. The wake half-width on both inner side and outer side of the wake decrease with mainstream velocity, whereas the maximum velocity defect, turbulence stresses increase with mainstream velocity. The turbulence stresses are enhanced on the inner side but suppressed on the outer side.     

LES of turbulent flow in a concentric annulus with rotating outer wall

Fully-developed turbulent flow in a concentric annulus, r₁/r₂ = 0.5, Re_h = 12,500, with the outer wall rotating at a range of rotation rates N = U_θ,wall/U_b from 0.5 up to 4 is studied by large-eddy simulations. The focus is on the effects of moderate to very high rotation rates on the mean flow, turbulence statistics and eddy structure. For N up to ∼2, an increase in the rotation rate dampens progressively the turbulence near the rotating outer wall, while affecting only mildly the inner-wall region. At higher rotation rates this trend is reversed: for N = 2.8 close to the inner wall turbulence is dramatically reduced while the outer wall region remains turbulent with discernible helical vortices as the dominant turbulent structure. The turbulence parameters and eddy structures differ significantly for N = 2 and 2.8. This switch is attributed to the centrifuged turbulence (generated near the inner wall) prevailing over the axial inertial force as well as over the counteracting laminarizing effects of the rotating outer wall. At still higher rotation, N = 4, the flow gets laminarized but with distinct spiralling vortices akin to the Taylor–Couette rolls found between the two counter-rotating cylinders without axial flow, which is the limiting case when N approaches to infinity. The ratio of the centrifugal to axial inertial forces, Ta/Re² ∝ N² (where Ta is the Taylor number) is considered as a possible criterion for defining the conditions for the above regime change.     

Convective heat transfer to CO2 at a supercritical pressure flowing vertically upward in tubes and an annular channel

The Super-Critical Water-Cooled Reactor (SCWR) has been chosen by the Generation IV International Forum as one of the candidates for the next generation nuclear reactors. Heat transfer to water from a fuel assembly may deteriorate at certain supercritical pressure flow conditions and its estimation at degraded conditions as well as in normal conditions is very important to the design of a safe and reliable reactor core. Extensive experiments on a heat transfer to a vertically upward flowing CO₂ at a supercritical pressure in tubes and an annular channel have been performed. The geometries of the test sections include tubes of an internal diameter (ID) of 4.4 and 9.0 mm and an annular channel (8 × 10 mm). The heat transfer coefficient (HTC) and Nusselt numbers were derived from the inner wall temperature converted by using the outer wall temperature measured by adhesive K-type thermocouples and a direct (tube) or indirect (annular channel) electric heating power. From the test results, a correlation, which covers both a deteriorated and a normal heat transfer regime, was developed. The developed correlation takes different forms in each interval divided by the value of parameter Bu. The parameter Bu (referred to as Bu hereafter), a function of the Grashof number, the Reynolds number and the Prandtl number, was introduced since it is known to be a controlling factor for the occurrence of a heat transfer deterioration due to a buoyancy effect. The developed correlation predicted the HTCs for water and HCFC-22 fairly well.     

The degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary

This paper investigates the degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary. Inside the boundary, there are many voids with arbitrary configurations. The problem is analyzed with a relevant homogenous BIE (boundary integral equation). It is assumed that all the inner void boundary tractions are equal to zero, and tractions on the outer circular boundary are constant. Therefore, all the integrations in BIE are performed on the outer circular boundary only. By using the relation z * conjg(z) = a * a, or conjg(z) = a * a/z on the circular boundary with radius a, all integrals can be reduced to an integral for complex variable and they can be integrated in closed form. The degenerate scale a = 1 is found in the Laplace equation and in plane elasticity regardless of the void configuration.     

Numerical and experimental study of an axially induced swirling pipe flow

In-line flow segregators based on axial induction of swirling flow have important applications in chemical, process and petroleum production industries. In the later, the segregation of gas bubbles and/or water droplets dispersed into viscous oil by swirling pipe flow may be beneficial by either providing a pre-separation mechanism (bubble and/or drop coalescer) or, in the case of water-in-oil dispersions, by causing a water-lubricated flow pattern to establish in the pipe (friction reduction). Works addressing these applications are rare in the literature. In this paper, the features and capabilities of swirling pipe flow axially induced by a vane-type swirl generator were investigated both numerically and experimentally. The numerical analysis has been carried out using a commercial CFD package for axial Reynolds numbers less than 2000. Pressure drop, tangential and axial velocity components as well as swirl intensity along a 5 cm i.d. size and 3 m long pipe were computed. Single phase flow experiments have been performed using a water–glycerin solution of 54 mPa s viscosity and 1210 kg/m³ density as working fluid. The numerical predictions of the pressure drop were compared with the experimental data and agreement could be observed within the range of experimental conditions. The experiments confirmed that swirl flow leads to much higher friction factors compared with theoretical values for non-swirl (i.e. purely axial) flow. Furthermore, the addition of a conical trailing edge reduces vortex breakdown. Visualization of the two-phase swirling flow pattern was achieved by adding different amounts of air to the water–glycerin solution upstream the swirl generator.     

Postbuckling prediction of double-walled carbon nanotubes under axial compression

An elastic double-shell model is presented for the buckling and postbuckling of a double-walled carbon nanotube subjected to axial compression. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual elastic shell and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. The van der Waals interaction between the inner and outer nanotubes and the nonlinear prebuckling deformations of the shell are both taken into account. A boundary layer theory of shell buckling is extended to the case of double-walled carbon nanotubes under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the single-walled carbon nanotube and the double-walled carbon nanotube both have an unstable postbuckling behavior.     

Developing structure of two-phase flow in a large diameter pipe at low liquid flow rate

In order to develop the interfacial area transport equation for the interfacial transfer terms in the two-fluid model, accurate data sets on axial development of local parameters such as void fraction, interfacial area concentration, interfacial gas velocity and Sauter mean diameter are indispensable to verify the modeled source and sink terms in the interfacial area transport equation. From this point of view, local measurements of both group 1 spherical/distorted bubbles and group 2 cap/slug bubbles in vertical upward air–water two-phase flow in a large diameter pipe with 200 mm in inner diameter and 26 m in height were performed at three axial locations of z/D = 41.5, 82.8 and 113 as well as 11 radial locations from r/R = 0–0.95 by using four-sensor probe method. Here, z, r, D and R are the axial distance from the inlet, radial distance from the pipe center, pipe diameter and pipe radius, respectively. The liquid flow rate and the void fraction ranged from 0.0505 m/s to 0.312 m/s and from 1.98% to 32.6%, respectively in the present experiment. The flow condition covered extensive region of bubbly flow, cap turbulent flow as well as their transition. The extensive analysis on the radial profiles of local flow parameters and their axial developments demonstrate the development of interfacial structures along the flow direction due to the bubble coalescence and breakup and the gas expansion. The significant decrease in void faction and interfacial area concentration and the increase in Sauter mean diameter and interfacial velocity were observed when the gradual flow regime transition occurred. Finally, the net change in the interfacial area concentration due to the bubble coalescence and breakup was quantitatively investigated in the present paper to reflect the true transfer mechanisms in observed two-phase flows.     

Effects of layer number and initial pressure on continuum-based buckling analysis of multi-walled carbon nanotubes accounting for van der Waals interaction

The structural instability of multi-walled carbon nanotubes (MWCNTs) has captured extensive attention due to the unique characteristic of extremely thin hollow cylinder structure. The previous studies usually focus on the buckling behavior without considering the effects of the wall number and initial pressure. In this paper, the axial buckling behavior of MWCNTs with the length-to-outermost radius ratio less than 20 is investigated within the framework of the Donnell shell theory. The governing equations for the infinitesimal buckling of MWCNTs are established, accounting for the van der Waals (vdW) interaction between layers. The effects of the wall number, initial pressure prior to buckling, and aspect ratio on the critical buckling mode, buckling load, and buckling strain are discussed, respectively. Specially, the four-walled and twenty-walled CNTs are studied in detail, indicating the fact that the buckling instability may occur in other layers besides the outermost layer. The obtained results extend the buckling analysis of the continuum-based model, and provide theoretical support for the application of CNTs.     

Buckling of embedded multi-walled carbon nanotubes under combined torsion and axial loading

This paper describes an investigation into elastic buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading, which takes account of the radial constraint from the surrounding elastic medium and van der Waals force between two adjacent tube walls. Depending on the ratio of radius to thickness, the multi-walled carbon nanotubes discussed here are classified as thin, thick, and nearly solid. Critical buckling load with the corresponding mode is obtained for multi-walled carbon nanotubes under combined torsion and axial loading, with various values of the radius to thickness ratio and surrounded with different elastic media. The study indicates that the buckling mode (m, n) of an embedded multi-walled carbon nanotube under combined torsion and axial loading is unique and it is different from that with axial compression only. New features for the buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading and the meaningful numerical results are useful in the design of nanodrive device, nanotorsional oscillator and rotational actuators, where multi-walled carbon nanotubes act as basic elements.     

Stratified flow model for convective condensation in an inclined tube

Experimental data are reported for condensation of R134a in an 8.38 mm inner diameter smooth tube in inclined orientations with a mass flux of 200 kg/m² s. Under these conditions, the flow is stratified and there is an optimum inclination angle, which leads to the highest heat transfer coefficient. There is a need for a model to better understand and predict the flow behaviour. In this paper, the state of the art of existing models of stratified two-phase flows in inclined tubes is presented, whereafter a new mechanistic model is proposed. The liquid–vapour distribution in the tube is determined by taking into account the gravitational and the capillary forces. The comparison between the experimental data and the model prediction showed a good agreement in terms of heat transfer coefficients and pressure drops. The effect of the interface curvature on the heat transfer coefficient has been quantified and has been found to be significant. The optimum inclination angle is due to a balance between an increase of the void fraction and an increase in the falling liquid film thickness when the tube is inclined downwards. The effect of the mass flux and the vapour quality on the optimum inclination angle has also been studied.