A criterion for detection of the onset of Dean instability in Newtonian fluids

Dean instability for Newtonian fluids in laminar secondary flow in 180° curved channels was studied experimentally and numerically. The numerical study used Fluent CFD code to solve the Navier–Stokes equations, focusing on flow development conditions and the parameters influencing Dean instability. An accurate criterion based on the radial gradient of the axial velocity was defined that allows detection of the instability threshold, and this criterion is used to optimize the grid geometry. The effects on Dean instability of the curvature ratio (from 5.5 to 20) and aspect ratio (from 0.5 to 12) are studied. In particular, we show that the critical value of the Dean number decreases with the increasing duct curvature ratio. The variation of the critical Dean number with duct aspect ratio is less regular.In the experimental study, flows were visualized in several tangential positions of a 180° curved channel with aspect ratio 8 and curvature ratio 10. The flow is hydrodynamically developed at the entrance to the curved channel. The critical Dean number is detected and the development of secondary flow vortices by additional counter-rotating vortex pairs is observed. A diagram of different critical Dean numbers is established.     

A finite element analysis of the steady laminar entrance flow in a 90° curved tube

A standard Galerkin finite element penalty function method is used to approximate the solution of the three-dimensional Navier–Stokes equations for steady incompressible Newtonian entrance flow in a 90° curved tube (curvature ratio δ = 1/6) for a triple of Dean numbers (κ = 41, 122 and 204). The computational results for the intermediate Dean number (κ = 122) are compared with the results of laser–Doppler velocity measurements in an equivalent experimental model. For both the axial and secondary velocity components, fair agreement between the computational and experimental results is found.     

Effect of magnetic field on the buoyancy and thermocapillary driven convection of an electrically conducting fluid in an annular enclosure

The main objective of this article is to study the effect of magnetic field on the combined buoyancy and surface tension driven convection in a cylindrical annular enclosure. In this study, the top surface of the annulus is assumed to be free, and the bottom wall is insulated, whereas the inner and the outer cylindrical walls are kept at hot and cold temperatures respectively. The governing equations of the flow system are numerically solved using an implicit finite difference technique. The numerical results for various governing parameters of the problem are discussed in terms of the streamlines, isotherms, Nusselt number and velocity profiles in the annuli. Our results reveal that, in tall cavities, the axial magnetic field suppresses the surface tension flow more effectively than the radial magnetic field, whereas, the radial magnetic field is found to be better for suppressing the buoyancy driven flow compared to axial magnetic field. However, the axial magnetic field is found to be effective in suppressing both the flows in shallow cavities. From the results, we also found that the surface tension effect is predominant in shallow cavities compared to the square and tall annulus. Further, the heat transfer rate increases with radii ratio, but decreases with the Hartmann number.     

An exact analytical solution for creeping Dean flow of Bingham plastics through curved rectangular ducts

In this paper, an exact analytical solution for creeping flow of Bingham plastic fluid passing through curved rectangular ducts is presented for the first time. The closed form of axial velocity distribution, flow resistance ratio, and wall shear stress are derived using bounded Fourier transformation. An extensive investigation on mutual effects of Hedstrom number, curvature ratio, and aspect ratio is conducted. The results indicate that a drag reduction is caused in the flow field by increasing the Hedstrom number. It is shown that unlike the Newtonian creeping Dean flow, the critical aspect ratio (an aspect ratio in which the flow resistance ratio is independent from curvature ratio) does not exist at large enough Hedstrom numbers. Analytical solution also indicated that as Hedstrom number is increased, the value of Poiseuille number is enhanced, and unlike the Newtonian flows, the value of Poiseuille number is not zero at edges of cross section.     

An experimental study of nonrheometric flows of viscoelastic fluids: II. Flow in a liquid jet exiting from a vertical tube

A laser anemometer has been used to study the region of accelerating shear flow near the exit of a vertical tube. It is in this region that the transition between steady laminar shear flow in the upstream tube and elongational flow in the downstream liquid jet takes place.Downstream velocity profiles were measured for solutions of 0.9% polyacrylamide in 85% glycerol/water and 0.9% polyacrylamide in water. Reynolds numbers (based on wall conditions in the fully developed upstream flow) ranged from 45 to 310 and Froude numbers from 0.294 to 4.11. Tubes, having sharpedged and rounded exit corners, with diameters of 1.25 cm and 1.90 cm were usedUpstream velocity profiles were measured for a solution of 0.9% polyacrylamide in water. Reynolds numbers ranged from 16 to 670. Only tubes having sharp-edged exit corners were used.It was found that the transition region did not extend upstream into the tube but was confined to the downstream jet. The transition took place over a distance of about 3–5 tube diameters depending upon the value of the Froude number. The axial distance downstream from the tube exit plane at which the velocity profile first became flat increased with increasing Froude number. The magnitude of the jet velocity at this point decreased with increasing Froude number.The condition of the tube exit corner was found to influence the flow in the transition region. Downstream velocity profiles obtained using tubes having rounded exit corners initially develop more slowly than, but soon catch up with and eventually overtake, the corresponding profiles obtained using tubes with sharp-edged exit corners.Downstream velocity profiles obtained for the 0.9% polyacrylamide in 85% glycerol/water solution were found to develop smoothly. The transition from steady shear flow in the tube to elongational flow in the jet took place through the combined processes of acceleration of the outer layers of the jet due to radial transfer of momentum with adjacent inner layers, the process spreading steadily inwards with increasing axial distance from the tube exit plane, and acceleration of the whole due to gravity. However, the velocity profiles obtained for the 0.9% polyacrylamide in water solution did not always develop so smoothly. At a Reynolds number of 310 and Froude number of 2.06 the radial momentum transfer process was restricted to a narrow outer region of the jet until a downstream axial distance of about 2 tube diameters was reached. Thereafter, the transition to a flat profile took place smoothly.     

Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with constant heat flux on the inner wall and a specified isothermal temperature on the outer wall. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. A physical model is introduced which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field are obtained for a range of modified Rayleigh number (10³<Ra?<5×10⁵), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (ε=±0.625). Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the modified Rayleigh number and the eccentricity of the spherical annulus. The results of average Nusselt numbers are also compared with the results obtained for flow between two isothermal spheres.     

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。     

A numerical‐variational procedure for laminar flow in curved square ducts

A new numerical method is presented for the solution of the Navier–Stokes and continuity equations governing the internal incompressible flows. The method denoted as the CVP method consists in the numerical solution of these equations in conjunction with three additional variational equations for the continuity, the vorticity and the pressure field, using a non‐staggered grid. The method is used for the study of the characteristics of the laminar fully developed flows in curved square ducts. Numerical results are presented for the effects of the flow parameters like the curvature, the Dean number and the stream pressure gradient on the velocity distributions, the friction factor and the appearance of a pair of vortices in addition to those of the familiar secondary flow. The accuracy of the method is discussed and the results are compared with those obtained by us, using a variation of the velocity–pressure linked equation methods denoted as the PLEM method and the results obtained by other methods. Copyright © 2004 John Wiley & Sons, Ltd.     

PLANAR LIQUID SHEETS AT LOW REYNOLDS NUMBERS

Asymptotic methods are employed to derive the leading-order equations which govern the fluid dynamics of time-dependent, incompressible, planar liquid sheets at low Reynolds numbers using as small parameter the slenderness ratio. Analytical and numerical solutions of relevance to both steady film casting processes and plane stagnation flows are obtained with the leading-order equations. It is shown that for steady film casting processes the model which accounts for both gravity and low-Reynolds-number effects predicts thicker and slower planar liquid sheets than those which neglect a surface curvature term or assume that Reynolds number is zero, because the neglect of the curvature term and the assumption of zero Reynolds number are not justified at high take-up velocities owing to the large velocity gradients that occur at the take-up point. It is also shown that for Reynolds number/Froude number ratios larger than one, models which neglect the surface curvature or assume a zero Reynolds number predict velocity profiles which are either concave or exhibit an inflection point, whereas the model which accounts for both curvature and low-Reynolds-number effects predicts convex velocity profiles. For plane stagnation flows it is shown that models which account for both low-Reynolds-number and curvature effects predict nearly identical results to those of models which assume zero Reynolds number. These two models also predict a faster thickening of the planar liquid sheet than models which account for low- Reynolds-number effects but neglect the surface curvature. This curvature term is very large near the stagnation point and cannot be neglected there. It is also shown that the thickening of the sheet occurs closer to the stagnation point as the Reynolds number/Froude number ratio is increased, i.e. as the magnitude of the gravitational acceleration is increased. In addition it is shown that large surface tension introduces a third-order spatial derivative in the axial momentum equation at leading order.     

Convective heat transfer for viscoelastic fluid in a curved pipe

In this paper, fully developed convective heat transfer of viscoelastic flow in a curved pipe under the constant heat flux at the wall is investigated analytically using a perturbation method. Here, the curvature ratio is used as the perturbation parameter and the Oldroyd-B model is applied as the constitutive equation. In the previous studies, the Dirichlet boundary condition for the temperature at the wall has been used to simplify the solution, but here exactly the non-homogenous Neumann boundary condition is considered to solve the problem. Based on this solution, the non-axisymmetric temperature distribution of Dean flow is obtained analytically and the effect of flow parameters on the flow field is investigated in detail. The current analytical results indicate that increasing the Weissenberg number, viscosity ratio, curvature ratio, and Prandtl number lead to the increase of the heat transfer in the Oldroyd-B fluid flow.     

Electrokinetic flow and energy conversion in a curved microtube

Curved channels are ubiquitous in microfluidic systems. The pressuredriven electrokinetic flow and energy conversion in a curved microtube are investigated analytically by using a perturbation analysis method under the assumptions of the small curvature ratio and the Reynolds number. The results indicate that the curvature of the microtube leads to a skewed pattern in the distribution of the electrical double layer (EDL) potential. The EDL potential at the outer side of the bend is larger than that at the inner side of the bend. The curvature shows an inhibitory effect on the magnitude of the streaming potential field induced by the pressure-driven flow. Since the spanwise pressure gradient is dominant over the inertial force, the resulting axial velocity profile is skewed into the inner region of the curved channel. Furthermore, the flow rate in a curved microtube could be larger than that in a straight one with the same pressure gradient and shape of cross section. The asymptotic solutions of the axial velocity and flow rate in the absence of the electrokinetic effect are in agreement with the classical results for low Reynolds number flows. Remarkably, the curved geometry could be beneficial to improving the electrokinetic energy conversion (EKEC) efficiency.     

Numerical simulation of the unsteady flow over an elliptic cylinder at different orientations

A numerical method is developed for investigating the two‐dimensional unsteady viscous flow over an inclined elliptic cylinder placed in a uniform stream of infinite extent. The direction of the free stream is normal to the cylinder axis and the flow field unsteadiness arises from two effects, the first is due to the flow field development following the start of the motion and the second is due to vortex shedding in the wake region. The time‐dependent flow is governed by the full conservation equations of mass and momentum with no boundary layer approximations. The parameters involved are the cylinder axis ratio, Reynolds number and the angle of attack. The investigation covers a Reynolds number range up to 5000. The minor–major axis ratio of the elliptic cylinder ranges between 0.5 and 0.6, and the angle of attack ranges between 0° and 90°. A series truncation method based on Fourier series is used to reduce the governing Navier–Stokes equations to two coupled infinite sets of second‐order differential equations. These equations are approximated by retaining only a finite number of terms and are then solved by approximating the derivatives using central differences. The results reveal an unusual phenomenon of negative lift occurring shortly after the start of motion. Various comparisons are made with previous theoretical and experimental results, including flow visualizations, to validate the solution methodology. Copyright © 2001 John Wiley & Sons, Ltd.     

Correlations for heat transfer coefficient and pressure drop in the annulus of concentric helical coils

Heat transfer and pressure drop characteristics in the annulus of concentric helical coils heat exchangers were experimentally investigated. The effects of coil curvature ratio, flow configuration, number of turns and addition of surfactant were investigated. Five test coils were designed and manufactured to study the effect of different parameters on heat transfer and pressure drop. The liquids used in the present study were water and oleyl-dihydroxy-etheyl-amine-oxide (ODEAO, C₂₂H₄₅NO₃ = 371) non-ionic aqua surfactant solution flowing through the annulus side. The inner side Reynolds number range 11,000–27,000 and the annulus side range 5,000–19,000. The results showed that the annulus Nusselt number decreases as annulus curvature ratio increases and increases when number of turns decrease. Moreover, the friction factor increases with the curvature ratio and also increases as number of turns decreases. Both Nusselt number and friction factor decrease when ODEAO concentration increases.     

Thermosolutal Convection in a Rectangular Concentric Annulus: A Comprehensive Study

This investigation presents numerical treatment of governing equations pertaining to thermosolutal flow within an annulus and an application of a model describing the important physical phenomenon as found in muffle furnace. The inner side of the annulus is exposed to high temperature and high solute concentration while the outer side of the annulus is maintained at low temperature and low solute concentration. Darcy-Brinkman-Forchheimer model is used to study the flow, heat and solute transfer in a non-Darcian saturated porous media. The solution is obtained upon application of control volume integration. Modified MAC method is used for the numerical solution of governing equations. Gradient dependent consistent hybrid upwind scheme of second order (GDCHUSSO) is used for discretization of the convective terms. The parameters such as Rayleigh-Darcy number, Darcy number, buoyancy ratio and width ratio, that govern the flow phenomenon have been identified and their effects are critically examined. The fluid flow pattern in the annular space and the associated heat and mass transfer are conceptualized from the obtained isoconcentration, isotherm and flowline contour maps.     

纳米粒子在方管中传输与沉降的研究

采用有限体积法离散并应用Simple方法对方截面弯曲管道内的纳米粒子传输和沉降进行了数值计算,结果表明Reynolds数和Schmidt数是影响纳米粒子传输和沉降的重要参数。粒子较小时,弯管中轴向速度较大的区域就是粒子的高浓度区域,沉降增强因子最大值出现在外弯侧的中心位置;粒子较大时,截面浓度的梯度值降低,沉降增强因子趋于平均,此时整个截面的粒子平均沉降。弯曲作用对于粒子较大且Dean数也较大时的影响更加明显。     

Bifurcation phenomena of generalized newtonian fluids in curved rectangular ducts

The bifurcation phenomenon in flow through a curved rectangular duct is investigated in this study. The non-linear equations of motion governing the steady, fully developed laminar flow of an incompressible generalized Newtonian fluid have been solved numerically. Extensive results have been generated in an effort to map the regions of multiple solution in the parameter space of Dean number, Dn, aspect ratio, γ, power-law index, n, and radius of curvature, r. For a Newtonian fluid (n = 1), at a fixed curvature (r = 100), the transition between a symmetric 2-cell and a symmetric 4-cell solution appears to follow a tilted cusp. The extent of the stable, symmetric 2-cell solution surface is critically influenced by the length scale γ. In the non-Newtonian case, at a fixed aspect ratio (γ = 1) and a fixed curvature (r = 100), the flow transition follows that of a fold catastrophe. The influence of the curvature is reasonably well accounted for in Dn. The bifurcation set determined in the Dn-γ space remains qualitatively the same at any value of n or r. These parameters merely shift and/or stretch the equilibrium surface determined by Dn and γ.     

The Dean instability in power-law and Bingham fluids in a curved rectangular duct

The laminar flow of power-law and yield-stress fluids in 180° curved channels of rectangular cross section was studied experimentally and numerically in order to understand the effect of rheological fluid behavior on the Dean instability that appears beyond a critical condition in the flow. This leads to the apparition of Dean vortices that differ from the two corner vortices created by the channel wall curvature.Flow visualizations showed that the Dean vortices develop first in the near-wall zone on the concave (outer) wall, where the shear rate is higher and the viscosity weaker; then they penetrate into the centre of the channel cross section where power-law fluids have high viscosity and Bingham fluids are unyielded in laminar flow. Based on the complete formation on the concave wall of the new pairs of counter-rotating vortices (Dean vortices), the critical value of the Dean number decreases as the power-law index increases for the power-law fluids, and the Bingham number decreases for the Bingham fluids. For power-law fluids, a diagram of critical Dean numbers, based on the number of Dean vortices formed, was established for different axial positions. For the same flow conditions, the critical Dean number obtained using the axial velocity gradient criterion was smaller then that obtained with the visualization technique.     

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 of second-order fluid in a curved duct with square cross-section

This paper investigates the inertial and creeping flow of a second-order fluid in a curved duct with a square cross-section. Numerical modeling is employed to analyze fluid flow, and the governing equations are discretized using the finite difference method on a staggered mesh. The marker-and-cell method is employed to allocate the parameters on the staggered mesh, and static pressure is calculated using the artificial compressibility approach. The effect of centrifugal force due to the curvature of the duct and the opposing effects of the first and second normal stress difference on the flow field are investigated. In addition, the order-of-magnitude technique is used to derive the force balance relations for the core region of flow. Based on these relations, the performance mechanism of centrifugal force and normal stress differences on the generation of secondary flows is considered. We also present an analytical relation for the axial velocity profile and flow resistance ratio of creeping flow. For this kind of flow, previous studies have investigated the effect of the first normal stress difference on the transition from one pair to two pairs of vortices while we show that the negative second normal stress difference has the opposite effect on this transition and can stabilize the flow.     

Numerical study on heavy rigid particle motion in a plane wake flow by spectral element method

Experimental particle dispersion patterns in a plane wake flow at a high Reynolds number have been predicted numerically by discrete vortex method (Phys. Fluids A 1992; 4 :2244–2251; Int. J. Multiphase Flow 2000; 26 :1583–1607). To address the particle motion at a moderate Reynolds number, spectral element method is employed to provide an instantaneous wake flow field for particle dynamics equations, which are solved to make a detail classification of the patterns in relation to the Stokes and Froude numbers. It is found that particle motion features only depend on the Stokes number at a high Froude number and depend on both numbers at a low Froude number. A ratio of the Stokes number to squared Froude number is introduced and threshold values of this parameter are evaluated that delineate the different regions of particle behavior. The parameter describes approximately the gravitational settling velocity divided by the characteristic velocity of wake flow. In order to present effects of particle density but preserve rigid sphere, hollow sphere particle dynamics in the plane wake flow is investigated. The evolution of hollow particle motion patterns for the increase of equivalent particle density corresponds to that of solid particle motion patterns for the decrease of particle size. Although the thresholds change a little, the parameter can still make a good qualitative classification of particle motion patterns as the inner diameter changes. Copyright © 2008 John Wiley & Sons, Ltd.