Numerical investigation of turbulent flow,heat transfer and entropy generation in a helical coiled tube with larger curvature ratio

Three-dimensional turbulent forced convective heat transfer and flow characteristics, and the non-dimensional entropy generation number in a helical coiled tube subjected to uniform wall temperature are simulated using the k–ε standard turbulence model. A finite volume method is employed to solve the governing equations. The effects of Reynolds number, curvature ratio, and coil pitch on the average friction factor and Nusselt number are discussed. The results presented in this paper cover a Reynolds number range of 2 × 10⁴ to 6 × 10⁴, a pitch range of 0.1–0.2 and a curvature ratio range of 0.1–0.3. The results show that the coil pitch, curvature ratio and Reynolds number have different effects on the average friction factor and Nusselt number at different cross-sections. In addition, the flow and heat transfer characteristics in a helical coiled tube with a larger curvature ratio for turbulent flow are different from that of smaller curvature ratio for laminar and turbulent flow in certain ways. Some new features that are not obtained in previous researches are revealed. Moreover, the effects of Reynolds number, curvature ratio, and coil pitch on the non-dimensional entropy generation number of turbulent forced convection in a helical coiled tube are also discussed.     

Flow visualization and LDV measurements of laminar flow in a helical square duct with finite pitch

Flow visualization and LDV measurements are performed on laminar flow in a helical square duct with finite pitch. The experimental observations are compared to results of numerical calculations employing the finite-volume method and assuming a fully developed flow. Good agreement is found between measured and computed velocity profiles. This suggests that the physical velocity components used in the numerical calculations are suitable to describe the flow. It is further demonstrated that the contravariant velocity components employed by some authors may lead to results that are difficult to interpret. Two stable solution branches are detected in the numerical calculations. For Reynolds numbers between the stable branches, unsteady and fully developed computations predict an oscillating flow between a two-vortex and a four-vortex structure. In the experiments, the flow normally retained a stable two-vortex structure in the numerically predicted unstable regime. However, by disturbing the flow at the duct inlet, a four-vortex flow that showed similarities to the computed flow could occasionally be obtained. For Reynolds numbers above 600, unsteady flow behavior was observed both experimentally and numerically, which might be an early sign of transition. In the experiments, Gdrtler-like extra vortices emerged spontaneously from the outer wall without disturbing the flow at the inlet. The same phenomenon was observed in the numerical calculations, assuming an unsteady and fully developed flow, but the extra vortices appeared with a lower frequency than in the experiments.     

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.     

Experimental study of heat transfer characteristics due to confined impinging two-dimensional jets

Experimental results are presented for characteristics of impingement heat transfer caused by three slot jets. Experimental values were obtained for the dimensionless distance H = 0.5−3, dimensionless pitch P = 6−16, and Reynolds number Re = 500−8000. For laminar impinging flow, they were compared with numerical results. For turbulent impinging flow, two peaks of the local Nusselt number were obtained behind the second nozzle. The position of the second peak approached the nozzle as the space between nozzle and impinged surface decreased. The average Nusselt number between the central and second nozzles was determined from the ratio P/H and the Reynolds number based on the pitch of the nozzles.     

Fluid flow in a helical pipe

Without simplifying the N-S equations of Germano's^[5], we study the flow in a helical circular pipe employing perturbation method. A third perturbation solution is fully presented. The first- second- and third-order effects of curvature κ and torsion τ on the secondary flow and axial velocity are discussed in detail. The first-order effect of curvature is to form two counter-rotating cells of the secondary flow and to push the maximum axial velocity to the outer bend. The two cells are pushed to the outer bend by the pure second-order effect of curvature. The combined higher-order (second-, third-) effects of curvature and torsion, are found to be an enlargement of the lower vortex of the secondary flow at expense of the upper one and a clockwise shift of the centers of the secondary vortices and the location of maximum axial velocity. When the axial pressure gradient is small enough or the torsion is sufficiently larger than the curvature, the location of the maximal axial velocity is near the inner bend. The equation of the volume flux is obtained from integrating the perturbation solutions of axial velocity. From the equation the validity range of the perturbation solutions in this paper can be obtained and the conclusion that the three terms of torsion have no effect on the volume flux can easily be drawn. When the axial pressure gradient is less than 22.67, the volume flux in a helical pipe is larger than that in a straight pipe.     

Convective mass transfer in helical pipes: effect of curvature and torsion

A 3D numerical analysis of the flow and mass transfer in helical pipes is presented. The interpretation of the flow patterns and their impact on mass transfer is shown to require a non-orthogonal pseudo-stream function based visualization. The strong coupling between torsion and curvature effects, and the resulting secondary flow regimes are well characterized by a parameter combining both the Dean (Dn) and Germano numbers (Gn). For membrane separation applications, helical modules combining high curvature with low torsion would alleviate concentration polarization and yield appreciable flux improvement.     

Laser-Doppler measurements of laminar and turbulent flow in a pipe bend

Laser-Doppler measurements are reported for laminar and turbulent flow through a 90° bend of circular cross-section with mean radius of curvature equal to 2.8 times the diameter. The measurements were made in cross-stream planes 0.58 diameters upstream of the bend inlet plane, in 30, 60 and 75° planes in the bend and in planes one and six diameters downstream of the exit plane. Three sets of data were obtained: for laminar flow at Reynolds numbers of 500 and 1093 and for turbulent flow at the maximum obtainable Reynolds number of 43 000. The results show the development of strong pressure-driven secondary flows in the form of a pair of counter-rotating vortices in the streamwise direction. The strength and character of the secondary flows were found to depend on the thickness and nature of the inlet boundary layers, inlet conditions which could not be varied independently of Reynolds number. The quantitative anemometer measurements are supported by flow visualization studies. Refractive index matching at the fluid-wall interface was not used; the measurements consist, therefore, of streamwise components of mean and fluctuating velocities only, supplemented by wall pressure measurements for the turbulent flow. The displacement of the laser measurement volume due to refraction is allowed for in simple geometrical calculations. The results are intenden for use as benchmark data for calibrating flow calculation methods.     

A numerical study of bifurcation in laminar flow in curved ducts

The bifurcation phenomenon whereby multiple-vortex secondary flow occurs in place of the normal two-vortex flow in laminar flow in curved ducts has previously been studied numerically by several researchers. However, the various results have been conflicting on many points. The present paper describes a set of numerical experiments conducted to study the effect of numerical accuracy on the solution. The results show that the transition from two- to four-vortex structure depends strongly on the differencing scheme and to a lesser extent on the grid size. The study also shows that as the Reynolds number of the flow increases, a two-vortex structure is re-established via a path which involves strongly asymmetric secondary flow patterns. These results are in agreement, at least qualitatively, with recent experimental theoretical and numerical results.     

A numerical study of fully developed laminar flows in pipes with two planar curvatures

The spectral element method is applied on unstructured tetrahedral elements to solve the Navier–Stokes equations for fully developed laminar flow in pipes with two planar curvatures. Specific implementations of the spectral element method to double curved pipes and parallelization are described. Previous studies on flows in pipes focused on constant curvature or torsion geometries, as well as pipes with varying curvature. This study focuses on the periodic variation of both the curvature as well as torsion by analysing a pipe having two planar curvatures. The effects of the three parameters defining the pipe are studied to isolate the curvature and torsion effect on the magnitude and angle of the secondary flow. Furthermore, the geometric effects on the wall shear stress are studied, as it is an important fluid flow property, especially in blood flows. Copyright © 2006 John Wiley & Sons, Ltd.     

Spiral vortices between concentric cylinders

Spiral vortices appearing in Couette-Taylor flows are studied by means of numerical simulation. Transition curves from Couette to spiral vortices for different radius ratios and wavenumbers have been calculated in order to test our technique. Critical Reynolds numbers, angular velocities and slopes of the spirals at the onset of the instability agree with previous results [1]. Non-linear solutions obtained by a pseudospectral collocation method are studied, and they show a weak net axial flow. In order to counteract this effect, which is absent in the usual experimental set-up, an axial pressure gradient has been included. This procedure has proved to be sufficient to make the axial flow negligible. The onset of a quasiperiodic flow for larger Reynolds numbers, corresponding to a secondary bifurcation is also presented.     

Entropy generation analysis of fully developed laminar forced convection in a helical tube with uniform wall temperature

In the present study, fully developed laminar flow and heat transfer in a helically coiled tube with uniform wall temperature have been investigated analytically. Expressions involving relevant variables for entropy generation rate contributed to heat transfer and friction loss, and total entropy generation rate have been derived. The effect of various flow and coil parameters like Reynolds number, curvature ratio, coil pitch, etc. on the entropy generation rate has been studied for two fluids- air and water. The results of the present study have been compared to the corresponding entropy generation values of straight pipe. Investigating the results, some optimum values for Reynolds number have been proposed and compared with the optimum Reynolds numbers of laminar flow inside a coiled tube subjected to constant heat flux boundary condition.     

亚临界雷诺数下圆柱绕流的大涡模拟

本文应用Smagorinsky涡粘性模式和二阶精度的有限体积法对圆柱绕流的流场进行大涡模拟．求解了非正交曲线坐标系下的N-S方程,对雷诺数为100和20000的工况进行了计算．计算结果与实验及动力涡粘性模式的结果进行了比较,表明计算对于层流及高亚临界雷诺数的湍流流动是合理的     

环形截面螺旋管道内二次流动特性的研究

从曲线柱坐标系下的N-S方程出发,以曲率和挠率为小参数,采用摄动法求解了环形截面螺旋管道内的黏性流动,给出了完全二阶摄动解,结果表明：当挠率为零时,二次流表现为上下对称的四个涡;当挠率不为零,涡的对称性遭到破坏,二次涡的强度和个数受De数和环形截面内外径之比δ的影响,轴向速度最大值在De数较小时靠近管道的内侧,随着De数的增加,其最大值向外侧移动。     

Numerical study on mixed convection from a constant wall temperature circular cylinder in zero-mean velocity oscillating cooling flows

Fluid flow and heat transfer of mixed convection from a constant wall temperature circular cylinder in zero-mean velocity oscillating cooling flows have been simulated based on the projection method with two dimensional exponential stretched staggered cylindrical meshes. Cycle mean temperature and secondary streaming are obtained by the method of partial sums of the Fourier series. Present numerical results are validated by comparing the heat transfer results of free convection and the secondary streaming of pure oscillating flow over a circular cylinder to published experimental and numerical results. The complete structures of the cycle mean temperature and secondary streaming patterns are provided by numerical simulations over wide ranges of the Reynolds number, the Keulegan–Carpenter number and the Richardson number. Based on turning points of the curves of the overall Nusselt numbers versus Reynolds numbers and the characteristics of the cycle averaged temperature and flow patterns, the heat transfer can be divided into three linear regimes (conduction, laminar convection, and turbulent convection dominated regimes) and two non-linear transition regimes. The effects of wave directions, amplitudes, frequencies, and buoyancy forces on the enhancement of heat transfer are also investigated. The effective ranges of the governing parameters for heat transfer enhancement are identified.     

A numerical study of heat and momentum transfer for tube bundles in crossflow

A numerical scheme is developed to predict the heat transfer and pressure drop coefficients in flow through rigid tube bundles. The scheme uses the Galerkin finite element technique. The conservation equations for laminar steady-state flow are cast in the form of streamfunction and vorticity equations. A Picard iteration method is used for the solution of the resulting system of non-linear algebraic equations. Results for the heat transfer and pressure drop coefficients are obtained for tube arrays of pitch ratios of 1·5 and 2·0. Very good agreement of the present results and experimental data obtained in the past is observed up to Reynolds numbers of 1000. It is also observed that the results of the present method show better agreement with the experimental data and that they are applicable for higher Reynolds numbers than results of other studies.     

Effect of the pattern of initial perturbations on the steady pipe flow regime

The influence of the inlet flow formation mode on the steady flow regime in a circular pipe has been investigated experimentally. For a given inlet flow formation mode the Reynolds number Re* at which the transition from laminar to turbulent steady flow occurred was determined. With decrease in the Reynolds number the difference between the resistance coefficients for laminar and turbulent flows decreases. At a Reynolds number approximately equal to 1000 the resistance coefficients calculated from the Hagen-Poiseuille formula for laminar steady flow and from the Prandtl formula for turbulent steady flow are equal. Therefore, we may assume that at Re > 1000 steady pipe flow can only be laminar and in this case it is meaningless to speak of a transition from one steady pipe flow regime to the other. The previously published results [1–9] show that the Reynolds number at which laminar goes over into turbulent steady flow decreases with increase in the intensity of the inlet pulsations. However, at the highest inlet pulsation intensities realized experimentally, turbulent flow was observed only at Reynolds numbers higher than a certain value, which in different experiments varied over the range 1900–2320 [10]. In spite of this scatter, it has been assumed that in the experiments a so-called lower critical Reynolds number was determined, such that at higher Reynolds numbers turbulent flow can be observed and at lower Reynolds numbers for any inlet perturbations only steady laminar flow can be realized. In contrast to the lower critical Reynolds number, the Re* values obtained in the present study, were determined for given (not arbitrary) inlet flow formation modes. In this study, it is experimentally shown that the Re* values depend not only on the pipe inlet pulsation intensity but also on the pulsation flow pattern. This result suggests that in the previous experiments the Re* values were determined and that their scatter is related with the different pulsation flow patterns at the pipe inlet. The experimental data so far obtained are insufficient either to determine the lower critical Reynolds number or even to assert that this number exists for a pipe at all.     

A combined analytical,numerical, and experimental study of shape-memory-alloy helical springs

In this paper, the pseudoelastic response of shape memory alloy (SMA) helical springs under axial force is studied both analytically and numerically. In the analytical solution two different approximations are considered. In the first approximation, both the curvature and pitch effects are assumed to be negligible. This is the case for helical springs with large ratios of mean coil radius to the cross sectional radius (spring index) and small pitch angles. Using this assumption, analysis of the helical spring is reduced to that of the pure torsion of a straight bar with circular cross section. A three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is reduced to the one-dimensional pure shear case and a closed-form solution for torsional response of SMA bars in loading and unloading is obtained. In the next step, the curvature effect is included and the SMA helical spring is analyzed using the exact solution presented for torsion of curved SMA bars. In this refined solution, the effect of the direct shear force is also considered. In the numerical analyses, the three-dimensional constitutive equations are implemented in a finite element method and using solid elements the loading–unloading of an SMA helical spring is simulated. Analytical and numerical results are compared and it is shown that the solution based on the SMA curved bar torsion gives an accurate stress analysis in the cross section of the helical SMA spring in addition to the global load–deflection response. All the results are compared with experimental data for a Nitinol helical spring. Several case studies are presented using the proposed analytical and numerical solutions and the effect of changing different parameters such as the material properties and temperature on the loading–unloading hysteretic response of SMA helical springs is studied. Finally, some practical recommendations are given for improving the performance of SMA helical springs used as energy dissipating devices, for example for seismic applications.     

Experimental study on the confined 3D laminar flow past a square prism with a high blockage ratio

An experimental PIV study is presented that addresses the confined 3D laminar flow behaviour past a square prism. The Reynolds number (Re), based on prism cross-section height varies between 100 and 256. The channel aspect ratio is 1/1 and the blockage ratio is 1/2.5. This geometry is representative of a passive method to enhance mixing in otherwise highly ordered laminar channel flow. It is found that the lateral walls exert a strong effect on the flow behaviour with two main consequences: (a) the onset of vortex shedding is delayed to a Re in the vicinity of 170, as opposed to the unconfined case where the critical Re is reported to be between 50 and 60 and (b) transition from the steady closed recirculation bubble regime to the vortex shedding regime is not abrupt. In particular, there is a range of Re for which the closed recirculation bubble pulsates with increasing amplitude prior to the onset of the Karman street regime. The experimental results are supported by numerical computations, and the relation of the results with the practical design of engineering systems is also discussed.     

旋转方形截面螺旋管内流动与传热特性

利用数值计算方法研究了旋转矩形截面螺旋管内的粘性流动，分析了在离心力，科氏力共同作用下曲线管道中的二次流动结构、轴向流速分布、截面温度分布、摩擦系数比以及管道Nusselt数比随各参数的变化情况。计算结果表明：当旋转方向和主流方向相同时，旋转的作用与增大Dean数的作用相同，使得管道摩擦系数变大，管道换热效果增强，而当旋转方向和主流方向相反时，管道内流动结构变化十分明显，当F≈-1．2时(F为科氏力与离心力之比)，二次流出现类似于直扭管内的鞍状流动结构，轴向速度类似于静止直管内的流动结构，管道内的摩擦系数与静止直管内的摩擦系数大约相等，换热效果减至最弱；挠率对流动结构以及摩擦系数比和Nusselt系数比的影响效果与F有关。     

Analysis of pipe flow transition. Part I. Direct numerical simulation

We have developed an accurate hybrid finite-difference code for the simulation of unsteady incompressible pipe flow. The numerical scheme uses compact finite differences of at least eighth-order accuracy for the axial coordinate, and Chebyshev and Fourier polynomials for the radial and azimuthal coordinates, respectively. Boundary conditions for the incompressible flow are enforced using an influence-matrix technique, and the Poisson equation for pressure is solved using a fast direct method. The code has been used to simulate and analyze the spatial transition process in developed laminar pipe flow at a Reynolds number of Re=2350. Results of the simulation are compared to experimental data given by Han, Tumin and Wygnanski [18]. PACS 47.11.+j, 47.20.Ft, 47.27.Cn