Steady natural convection flow of a particulate suspension through a circular pipe

A continuum model for two-phase (fluid/particle) flow induced by natural convection is developed and applied to the problem of steady natural convention flow of a particulate suspension through an infinitely long pipe. The wall of the pipe is maintained at a constant temperature. The particle phase is endowed by an artificial viscosity which may be used to model particle-particle interaction in dension suspensions. Boundary conditions borrowed from rarefied gas dynamics are employed for the particle-phase wall conditions. Closed-form solutions for the velocity and temperature profiles are obtained. For the assumptions employed in the problem, the temperatures of both phases in the pipe are predicted to be uniform. A parametric study of some physical parameters involved in the problem is performed to illustrate the influence of these parameters on the velocity profiles of both the fluid and particle phases.     

用三维半人工瞬变法计算圆管内圆柱绕流

本文提出一种计算三维定常流动的半人工瞬变法,本方法的特点是直接利用流体力学的原始基本方程组进行数值计算。运动方程中的一个分量方程被用于计算压力,另外两个分量方程被加入人工瞬变项而成为人工瞬变方程,这两个人工瞬变方程被用于计算速度的两个分量,第三个速度分量则通过连续性方程进行计算得到。根据半人工瞬变方程组的特点和流动区域的特性,本方法采用半交错不等距非正交曲线贴体混合网格系进行数值计算,并利用质点导数差分格式使计算更简便。本文以圆管中不可压缩流体对圆柱的三维定常绕流问题为算例,具体画出计算用的半交错不等距非正交曲线贴体混合网格系,介绍三维半人工瞬变法的计算方法和步骤,并通过数值计算得到了此算例的计算结果。     

Flow pattern transition,pressure gradient,hold-up predictions in gas/non-Newtonian power-law fluid stratified flow

This work focuses on gas/non-Newtonian power-law fluid stratified pipe flow. Two different theoretical approaches to obtain pressure gradient and hold-up predictions are presented: the steady fully developed two-fluid model and the pre-integrated model. The theoretical predictions are compared with experimental data available for horizontal and for slightly downward inclined air/shear thinning fluid stratified flow taken from literature. The predictions of the pre-integrated model are validated showing a good agreement when compared with experimental data. The criteria for the transition from the stratified flow pattern are applied to gas/non-Newtonian stratified flow. The neutral stability analysis (smooth/wavy stratified flow) and the well-posedness (existence region of stratified flow) of governing equations are carry out. The predicted transition boundaries are obtained using the steady fully developed two-fluid model and the pre-integrated model, where the shape factors and their derivatives are accounted for. A comparison between the predicted boundaries and experimental flow pattern maps is presented and shows a good agreement. A comment on the shear stress modeling by the pre-integrated model is provided.     

Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder

The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analyt- ically using the homotopy analysis method （HAM）. Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numericM and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.     

Magnetohydrodynamic pipe flow in ducts with partial circular ring cross section and annular cross section

In this paper we use the Green function method to solve the problem of steady one dimensional flow of an incompressible viscous, electrically conducting fluid through a pipe with partial circular ring cross section and one with annular cross section, in the presence of an applied transverse uniform magnetic field. We obtain analytic solutions and carry out some numerical calculations of the velocity distribution and induced magnetic field.     

流体黏性对输流管道运动方程及临界流速的影响

考虑实际流体黏性引起的管内流速非均匀分布,针对层流和两种不同的湍流流态,对理想流体情况下输流管道运动方程中的离心力项进行了修正,得到的修正系数分别为1.333(圆管层流)、1.020(光滑管壁圆管湍流)和1.037～1.055(粗糙管壁圆管湍流).根据修正后的运动方程得到的上述3种情况下的发散失稳临界流速比理想流体流动情况下依次分别低13.4%,1.0%和1.8%～2.6%.流体黏性对输流管道运动方程及临界流速的影响只与流态有关,雷诺数决定流态,而黏性系数通过雷诺数间接起作用.     

Wave propagation in a fluid flowing through a curved thin-walled elastic tube

The pulsatile flow in a curved elastic pipe of circular cross section is investigated. The unsteady flow of a viscous fluid and the wall motion equations are written in a toroidal coordinate system, superimposed and linearized over a steady state solution. Being the main application relative to the vascular system, the radius of the pipe is assumed small compared with the radius of curvature. This allows an asymptotic analysis over the curvature parameter. The model results an extension of the Womersley's model for the straight elastic tube. A numerical solution is found for the first order approximation and computational results are finally presented, demonstrating the role of curvature in the wave propagation and in the development of a secondary flow.     

Resonance behaviour of viscoelastic fluid in Poiseuille flow in the presence of a transversal magnetic field

The oscillatory flow of a viscoelastic fluid in a circular pipe under the influence of a transversal magnetic field is studied. Exact solutions for the axial velocity and flow rate are presented. The velocity enhancement and the resonance behaviour are analysed both numerically and asymptotically in the case of small pipe radii. Approximations for the resonance frequencies and the achievable velocity enhancements are derived. Copyright © 2005 John Wiley & Sons, Ltd.     

A representation of the solution of the Gromeko problem for viscous and elastico-viscous fluids

The solution of the Gromeko problem [1] on unsteady flow of a viscous fluid in a long circular pipe is among the few exact solutions of the Navier-Stokes equations. Its effective solution is obtained only when the longitudinal pressure gradient is given as an arbitrary time function. However, in practice we encounter cases when the flow rate is a known time function. This sort of problem arises, in particular, in rheological experiments using viscometers with a given flow rate. In this case the determination of the pressure gradient from the given flow rate leads in the general case to a very unwieldy expression. Below we present an effective solution of this problem for viscous and elasticoviscous media using the method of solving the inlet flow problem for a steady flow of a viscous fluid in a semi-infinite pipe. It is shown that for the case of a viscous fluid these two problems are actually equivalent.     

Rheological predictions of a transversely isotropic fluid model with extensible microstructure

A continuum extensible director theory was formulated to describe the isothermal, incompressible flow of uniaxial rodlike semiflexible liquid crystalline polymers. The model is strictly restricted to material that flow-align in shear, and that, in the absence of flow, are sufficiently far from the nematic-isotropic phase transition. The microstructure of the continuum is described by a variable length director, but the extensibility is finite. The model is an extension of the TIF (Transversely Isotropic Fluid) model of Ericksen (1960). The thermodynamic restrictions on the model parameters are found using the non-negative definiteness of the entropy production. The rheological material functions predicted by the model are calculated for steady simple shear and steady uniaxial extensional flows. In the rigid rod limit the model predictions agree with those of the TIF model, and for the finite extensibility case the model predictions are in agreement with those associated with flexible isotropic polymers: strong non-Newtonian shear viscosity, positive first normal stress differences, recoverable shear of order one, negative second normal stress differences, and a maximum in the steady uniaxial extensional viscosity.     

A three-dimensional non-linear constitutive law for magnetorheological fluids, with applications

In this paper we first summarize the magnetic and mechanical balance equations for magnetorheological fluids undergoing steady motion in the presence of a magnetic field. A general three-dimensional non-linear constitutive law for such a fluid is given for the case in which the magnetic induction vector is used as the independent magnetic variable. The equations are needed for the analysis of boundary-value problems involving fluids with dispersed micron-sized ferrous particles subjected to a time-independent magnetic field. For illustration, the equations are applied, in the case of an incompressible fluid, to the solution of some basic problems. We consider unidirectional flow in a region confined by two infinite parallel plates with a magnetic field applied perpendicular to the plates. Next, we examine two problems involving a circular cylindrical geometry with the fluid occupying the region between two concentric cylinders: axial flow subjected to an axial magnetic field and circumferential flow with a circumferential field. After making some simplifying assumptions on the constitutive law and choosing material parameters, numerical solutions for the velocity profiles are illustrated.     

Entropy analysis for third grade fluid flow with Vogel model viscosity in annular pipe

The steady-state flow of a third grade fluid between concentric circular cylinders is considered and entropy generation due to fluid friction and heat transfer in the annular pipe is examined. Depending upon the fluid viscosity, entropy generation in the flow system varies. The third grade fluid is employed to account for the non-Newtonian effect while Vogel model is accommodated for temperature-dependent viscosity. The analysis is based on perturbation technique. The closed form solutions for velocity, temperature and entropy fields are presented. Entropy generation due to fluid friction and heat transfer in the flow system is formulated. The influence of viscosity parameters A and B on the entropy generation number is investigated. It is found that entropy generation number reduces with increasing viscosity parameter A, which is more pronounced in the region close to the annular pipe inner wall and opposite is true for increasing viscosity parameter B.     

Numerical simulation of circulation in gas-liquid column reactors: isothermal, bubbly, laminar flow

The gas-liquid flow inside a circular, isothermal column reactor with a vertical axis has been studied using numerical simulations. The flow is assumed to be in the laminar, bubbly flow regime which is characterized by a suspension of discrete air bubbles in a continuous liquid phase such as glycerol water. The mathematical formulation is based on the conservation of mass and momentum principle for the liquid phase. The gas velocity distribution is calculated via an empirically prescribed relative velocity as a function of void fraction. The interface viscous drag forces are prescribed empirically. For some cases a profile shape is assumed for the void ratio distribution. The influence of various profile shapes is investigated. The results are compared with those where the void ratio distribution is calculated from the conservation of mass equation. The mathematical model has been implemented by modifying a readily available computer code for single-phase newtonian fluid flows. The numerical discretization is based on a finite volume approach. The predictions show a good agreement with measurements. The circulation pattern seems not to be so sensitive to the actual shape of the void fraction profiles, but the inlet distribution of it is important. A significantly different flow pattern results when the void fraction distribution is calculated from the transport equation, as compared to those with a priori prescribed profiles. When the void fraction is uniformly distributed over the whole distributor plate, no circulation is observed. Calculations also show that even the two-phase systems with a few discrete bubbles can be simulated successfully by a continuum model.     

On the performance of sudden‐expansion pipe without/with cross‐flow injection: experimental and numerical studies

The paper explores the possibilities that different turbulence closures offer, for in‐depth analysis of a complex flow. The case under investigation is steady, turbulent flow in a pipe with sudden expansion without/with normal‐to‐wall injection through jets. This is a typical geometry where generation of turbulence energy takes place, due to sudden change in boundary conditions. This study is aimed at investigating the capability of a developed computational program by the present authors with three different turbulence models to calculate the mean flow variables. Three two‐equation models are implemented, namely the standard linear k ? ε model, the low Reynolds number k ? ε model and the cubic nonlinear eddy viscosity (NLEV) k ? ε model. The performance of the chosen turbulence models is investigated with regard to the available data in the literature including velocity profiles, turbulent kinetic energy and reattachment position in a pipe expansion. In order to further assess the reliability of the turbulence models, an experimental program was conducted by the present authors also at the fluid mechanics laboratory of Menoufiya University. Preliminary measurements, including the surface pressure along the two walls of the expansion pipe and the pressure drop without and with the presence of different arrangements of wall jets produced by symmetrical or asymmetrical fluid cross‐flow injection, are introduced. The results of the present studies demonstrate the superiority of the cubic NLEV k ? ε model in predicting the flow characteristics over the entire domain. The simple low Reynolds number k ? ε model also gives good prediction, especially when the reattachment point is concerned. The evaluation of the reattachment point and the pressure‐loss coefficient is numerically addressed in the paper using the cubic NLEV k ? ε model. The results show that the injection location can control the performance of the pipe‐expansion system. It is concluded that the introduction of flow injection can increase the energy loss in the pipe expansion. The near‐field turbulence structure is also considered in the present study and it is noticed that the turbulence level is strongly affected by the cross‐flow injection and the jet location. Copyright © 2011 John Wiley & Sons, Ltd.     

Gas-particle two-phase turbulent flow in a vertical duct

Two-phase gas-phase turbulent flows at various loadings between the two vertical parallel plates are analyzed. A thermodynamically consistent turbulent two-phase flow model that accounts for the phase fluctuation energy transport and interaction is used. The governing equation of the gas-phase is upgraded to a two-equation low Reynolds number turbulence closure model that can be integrated directly to the wall. A no-slip boundary condition for the gas-phase and slip-boundary condition for the particulate phase are used. The computational model is first applied to dilute gas-particle turbulent flow between two parallel vertical walls. The predicted mean velocity and turbulence intensity profiles are compared with the experimental data of Tsuji et al. (1984) for vertical pipe flows, and good agreement is observed. Examples of additional flow properties such as the phasic fluctuation energy, phasic fluctuation energy production and dissipation, as well as interaction momentum and energy supply terms are also presented and discussed.

Applications to the relatively dense gas-particle turbulent flows in a vertical channel are also studied. The model predictions are compared with the experimental data of Miller & Gidaspow and reasonable agreement is observed. It is shown that flow behavior is strongly affected by the phasic fluctuation energy, and the momentum and energy transfer between the particulate and the fluid constituents.     

Very slow flow of Bingham viscoplastic fluid around a circular cylinder

Numerical simulations have been used to study the flow of a Bingham viscoplastic fluid around a circular cylinder in an infinite medium with negligible inertia effects. Papanastasiou's regularisation technique has been adopted to approximate the model. The case corresponding to preponderant plasticity effects has been particularly studied and convergence of the solutions examined in detail. The flow kinematics and stresses have been determined. The rigid zones have been identified and characterised. At large Oldroyd numbers, when plasticity effects become preponderant, a viscoplastic boundary layer appears around the cylinder. The characteristics of this viscoplastic boundary layer are quantified. The results are compared with existing theoretical results, concerning particularly the predictions of the viscoplastic boundary layer theory and the plasticity theory.     

幂律流体圆柱绕流的格子波尔兹曼模拟简

基于插值补充格子波尔兹曼方法和幂律流体的本构方程,建立了贴体坐标系下适用于幂律流体的格子波尔兹曼模型,模拟了幂律流体的圆柱绕流问题,采用非平衡外推格式处理圆柱表面的速度无滑移边界,利用应力积分法确定曳力系数和升力系数,并与基于标准的格子波尔兹曼方法和有限容积法获得的数值数据进行对比,吻合良好. 进行了网格无关性验证之后,分析了稳态流动时,不同雷诺数下幂律指数对于尾迹长度、分离角、圆柱表面黏度分布、表面压力系数及曳力系数的影响,以及非定常流动中,幂律指数对于流场、曳力系数、升力系数和斯特劳哈尔数的影响. 获得的变化规律与基于其他数值模拟方法得到的结果相一致,充分验证了模型的有效性和正确性. 结果表明：插值补充格子波尔兹曼方法可以用来模拟幂律流体在具有复杂边界流场内的流动问题,通过引入不同的非牛顿流体本构方程,该方法还可以进一步应用于其他类型的非牛顿流体研究中.     

O(2)‐symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

The steady, axisymmetric laminar flow of a Newtonian fluid past a centrally‐located sphere in a pipe first loses stability with increasing flow rate at a steady O(2)‐symmetry breaking bifurcation point. Using group theoretic results, a number of authors have suggested techniques for locating singularities in branches of solutions that are invariant with respect to the symmetries of an arbitrary group. These arguments are presented for the O(2)‐symmetry encountered here and their implementation for O(2)‐symmetric problems is discussed. In particular, how a bifurcation point may first be detected and then accurately located using an ‘extended system’ is described. Also shown is how to decide numerically if the bifurcating branch is subcritical or supercritical. The numerical solutions were obtained using the finite element code ENTWIFE. This has enabled the computation of the symmetry breaking bifurcation point for a range of sphere‐to‐pipe diameter ratios. A wire along the centerline of the pipe downstream of the sphere is also introduced, and its effect on the critical Reynolds number is shown to be small. Copyright © 2000 John Wiley & Sons, Ltd.     

Unsteady generalized Couette flow in composite microchannel

A numerical study is reported on the fully developed unsteady laminar fluid flow in microchannel parallel-plates partially filled with a uniform porous medium and partially filled with a clear fluid. The flow is induced by the movement of one of the plates and the pressure gradient. The Brinkman-extended Darcy model is utilized to model the flow in the porous region, while the Stokes equation is used in the clear fluid region. A theoretical analysis is also presented for the fully developed steady flow to find closedform expressions for the interfacial velocity and the velocity and skin frictions at the bounding plates. Numerical computations shows excellent agreement between the closedform solutions for fully developed steady flow and the numerical solution to unsteady flow at large values of time.     

Two‐phase pressure drop caused by sudden flow area contraction/expansion in small circular pipes

Theoretical studies have been made to determine the pressure drops caused by abrupt flow area expansion/contraction in small circular pipes for two‐phase flow of air and water mixtures at room temperature and near atmospheric pressure. Two‐phase computational fluid dynamics (CFD) calculations, using Eulerian–Eulerian model (with the air phase being compressible for pipe contraction case) are employed to calculate the pressure drop across sudden expansion and contraction. The pressure drop is determined by extrapolating the computed pressure profiles upstream and downstream of the expansion/contraction. The larger and smaller tube diameters are 1.6 and 0.84 mm, respectively. Computations have been performed with single‐phase water and air, and two‐phase mixtures in a range of Reynolds number (considering all‐liquid flow) from 1000 to 12 000 and flow quality from 1.2 × 10^?3 to 1.6 × 10^?2. The numerical results are validated against experimental data from the literature and are found to be in good agreement. The expansion and contraction loss coefficients are found to be different for single‐phase flow of air and water, and they agreed reasonably well with the commonly used theoretical predictions. Based on the numerical results as well as experimental data, correlations are developed for two‐phase flow pressure drops caused by the flow area contraction as well as expansion. Copyright © 2010 John Wiley & Sons, Ltd.