Viscous flow of a suspension of liquid drops in a cylindrical tube

Viscous flow in a circular cylindrical tube containing an infinite line of viscous liquid drops equally spaced along the tube axis is considered under the assumption that a surface tension, sufficiently large, holds the drops in a nearly spherical shape. Three cases are considered: (1) axial translation of the drops, (2) flow of the external fluid past a line of stationary drops, and (3) flow of external fluid and liquid drops under an imposed pressure gradient. Both fluids are taken to be Newtonian and incompressible, and the linearized equations of creeping flow are used.The results show that both drag and pressure drop per sphere increase as the spacing increases at fixed radius and also increase as the radius of the drop increases. The presence of the internal motion reduces the drag and pressure gradients in all cases compared to rigid spheres, particularly for drops approaching the size of the tube.     

Three-dimensional numerical simulation of sedimenting drops inside a vertical channel

Numerical simulation of sedimenting deformable drops inside a vertical channel has been performed at finite Reynolds numbers. The channel is confined by two vertical walls in x-direction and is periodic in y- and z-directions. Results are obtained using a finite difference/front-tracking method. The main dimensionless parameters are the Reynolds number, the Bond number and ratio of the length of channel to the diameter of drops. The effect of these parameters on lateral migration of drops is investigated. It is found that the wall repulsion is the main mechanism of the lateral migration of the drop, and drop migrates toward the channel axis. When the Reynolds number is relatively low, two different lateral migration regimes are observed: migration with monotonic approach and migration with damped oscillations. These regimes are affected by the dimensionless parameters. When the Bond number increases, the oscillations of drop around the centerline of channel are stronger and drop reaches the channel centerline in a larger period. Results of lateral migration of one drop are consistent with perturbation theory, and two-dimensional numerical simulations performed by Feng et al. (1994). The drag coefficient has also been calculated, and effect of various parameters has been discussed. Two drops interaction is similar to that observed by Feng et al. (1994) for two-dimensional circular cylinders. Results are consistent with experiments performed by Wu and Manasseh (1998). Simulations of four sedimenting drops show that depending on the relative size of drops, they either fall in two rows or they form a single horizontal layer and settle with a unique velocity.     

Stokes flow inside and around a fluid drop distorted by the presence of a tube wall

Using a quadratic optimisation process to satisfy the boundary conditions, the drag coefficient and the flow patterns inside and outside a fluid drop translating axially in a tube have been determined accurately even when the drop is clearly elongated. Convincing comparisons with experimental results (flow visualizations, velocity and drag measurements) are presented. The effects of the viscosity and density ratios are examined.     

Investigation on effect of drag models on flow behavior of power-law fluid–solid two-phase flow in fluidized bed

In this study, a Eulerian-Eulerian two-fluid model combined with the kinetic theory of granular flow is adopted to simulate power-law fluid–solid two-phase flow in the fluidized bed. Two new power-law liquid–solid drag models are proposed based on the rheological equation of power-law fluid and pressure drop. One called model A is a modified drag model considering tortuosity of flow channel and ratio of the throat to pore, and the other called model B is a blending drag model combining drag coefficients of high and low particle concentrations. Predictions are compared with experimental data measured by Lali et al., where the computed porosities from model B are closer to the measured data than other models. Furthermore, the predicted pressure drop rises as liquid velocity increases, while it decreases with the increase of particle size. Simulation results indicate that the increases of consistency coefficient and flow behavior index lead to the decrease of drag coefficient, and particle concentration, granular temperature, granular pressure, and granular viscosity go down accordingly.     

横向气流穿过垂直下落液滴场的二维流动数值计算

对Cl2/He混合气体横向穿过垂直下落的BHP(按重量25%的KOH,25%的H2O2及50%的H2O)液滴场的化学反应流动作了数值计算.模拟的流场是气体/液滴两相流场,在气相方程中,考虑了液滴与Cl2反应产生及释放O2(1Δ)的质量源项及表示液滴对气流阻碍作用的动量源项.由于气相动量小,液滴在下落过程横向偏移小,下落...     

Transmission coefficient of a grid electrode

The behavior of conducting emulsion drops (carrying a constant charge) in the region between electrodes connected to dc sources is studied. Assuming that the concentration of the dispersed phase is reasonably low, the problem may be reduced to one of determining the motion of an isolated drop close to the electrode. The trajectories of the drops in a flow passing around the electrode are then calculated, allowing for charge exchange between the drops and the electrode, and the electrode transmission coefficient is determined in relation to the parameters of the problem. An analogous situation was envisaged in earlier papers [1, 2] for a single cylindrical electrode but without allowing for the recharging of the particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 157–158, May–June, 1974.     

Effect of momentum transfer condition at the interface of a model of creeping flow past a spherical permeable aggregate

Creeping flow past an isolated, spherical and permeable aggregate has been studied adopting the Stokes equation to model the fluid external to the aggregate and the Brinkman equation for the internal flow. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used along with the continuity of velocity components and continuity of the normal stress. Using Faxen’s laws, drag and torque are calculated for different flow conditions and it is observed that drag and torque not only change with the permeability of the porous region, but as stress jump coefficient increases, the rate of change in behavior of drag and torque increases.     

Numerical investigation of different modes of internal circulation in spherical drops: Fluid dynamics and mass/heat transfer

The results of detailed, three-dimensional numerical simulations of fixed spherical drops in a uniform flow are presented. The fluid dynamics outside and inside of the drops as well as the internal problem of mass (or heat) transfer are studied. Liquid drops in both a liquid and a gaseous ambient phase are considered. Special emphasis is put on the investigation of different modes of internal circulation.At low Reynolds numbers of the inner fluid, the flow field inside the drop resembles the well known Hill’s vortex solution. However, at higher internal Reynolds numbers, stable steady or quasi-steady alternative modes of internal circulation are found. As these modes are not cylindrical symmetric around the streamwise axis, the often applied assumption of a two-dimensional, axisymmetric flow field is not justified in these cases. Thus, major discrepancies to previous numerical studies are obtained. However, it is shown that experimental results support our findings.For liquid drops surrounded by a liquid, a major influence of the state of internal circulation on the drag is discovered, whereas the drag is nearly unaltered in the case of a liquid drop in gas.Concerning the internal problem of mass/heat transfer, the various internal flow modes show different characteristics. At low internal Peclet numbers, higher Sherwood numbers are reached for the Hill’s vortex-like cases, whereas at higher Peclet numbers, the transfer is faster for the alternative modes. For cases with a Hill’s vortex-like solution, asymptotic Sherwood numbers for very high Peclet numbers of around 20 are found, whereas no upper limit for cases with alternative modes can be determined. In the present study a maximum internal Sherwood number of 130 is reached, more than six times the maximum value for a case with a Hill’s vortex-like internal solution.     

Interaction between two consecutive axisymmetric Taylor drops flowing in a heavier liquid in a vertical tube

A study regarding the interaction between two consecutive Taylor drops flowing in a heavier liquid in a vertical tube is reported. Under certain conditions, due to the wake of the leading drop, the trailing drop accelerates, leading to coalescence of the two drops. This study was developed using a numerical model based in the Volume of Fluid method in an axisymmetric geometry. The simulations reported in the present work had to fulfill two conditions: axisymmetry (due to the numerical model) and a high enough drop Reynolds number (which is related to the disturbances in the wake of an isolated drop, and thus to the tendency to drop interaction). Relevant dimensionless numbers are used to assess the effect of the acting forces. Detailed flow patterns and drop shapes are provided. Furthermore, the approaching velocity acquired by the trailing drop is analyzed and velocity profiles between the leading and the trailing drop are also reported. In general, the trailing drop shows an accelerating region, followed by a deceleration near the leading drop. The increase of Eotvos number promotes higher accelerations, while the increase in Morton number and viscosity ratio has the opposite effect. By comparison to literature gas-liquid studies, it was also found that interfacial forces promote the shape stability of the drops.     

Pipe flow with solid particles fixed in space

A group of solid particles were hung by slender rods in a pipe to make a model of two-phase flow of coarse particles. Pressure gradient and velocities were measured for different types of the models. The drag on the particles (spheres) were obtained from measurements of pressure gradient with some assumptions. The results are summarized as follows. (1) Mean velocities of fluid are lower in the central part of the pipe than in the circumferential part. Turbulence is remarkably increased by particles. The spectrum distribution of turbulent velocity becomes flatter. These results are similar to the gas-solid flow of coarse particles in a vertical pipe. (2) At a large Reynolds number, the drag coefficient per one sphere in the group is larger than that of a single isolated sphere in a uniform flow. When the spheres are arranged along the same line in the longitudinal direction, the drag coefficient becomes smaller as the longitudinal distance between the spheres is shortened.     

基于浸入边界-格子Boltzmann通量求解法的椭圆柱流动特性分析

基于浸入边界-格子Boltzmann通量求解法,开展了雷诺数Re=100不同几何参数下单椭圆柱及串列双椭圆柱绕流流场与受力特性对比研究。结果表明,随长短轴比值的增加,单椭圆柱绕流阻力系数先减小后缓慢上升,最大升力系数则随长短轴比值的增大而减小;尾迹流动状态从周期性脱落涡到稳定对称涡。间距是影响串列圆柱及椭圆柱流场流动状态的主要因素,间距较小时,串列圆柱绕流呈周期性脱落涡状态,而椭圆柱则为稳定流动;随着间距增加,上下游圆柱及椭圆柱尾迹均出现卡门涡街现象,且串列椭圆柱临界间距大于串列圆柱。串列椭圆柱阻力的变化规律与圆柱的基本相同,上游平均阻力大于下游阻力;上游椭圆柱阻力随着间距的变大先减小,下游随间距的变大而增加,当间距达到临界间距时上下游阻力跃升,随后出现小幅度波动再逐渐增加,并趋近于相同长短轴比值下单柱体绕流的阻力。     

热流体力学及其应用

热流体力学是一门涉及传热学、流体力学和热力学的交叉学科,并把重点放在讨论热过程对流体流动的影响。它由5部分组成:①热阻力。在某些情况下热阻力的存在对通道中的流体流量和换热系数有重大影响。借助于热阻力系数的定义和分析表达式,不仅可以预示单相通道流中的压力降,而且能用简便的方法预示气-液两相通道流中的压力降和临界热流。②热绕流。运用“虚质量源”和“热偶极子”的概念,对热绕流现象进行了分析和数值研究。它可在热除尘、粒子样品收集和热设备中流量分配等方面获得广泛的应用。③热驱动。不仅在重力场中,而且在如离心力场、表面张力场和电磁力场中也存在着热驱动流。着重讨论了流体运动的起因及其带来的后果,它包括环境污染、传热强化和同位素分离系数的提高等,④热不稳定性。重点讨论了热不稳定性的物理机理。用各种动力学方法所得到的流动不稳定性的临界准则对材料加工、热减阻、水源热污染等都是十分重要的。⑤热优化。研究了基于熵产生最小(热力学第二定律)为目标函数的流动和传热过程的优化。探讨了在一定条件下热力学第一定律效率和第二定律效率的内在联系。     

Heat and mass transfer in the vicinity of an evaporating droplet

The problem of heat and mass transfer around an evaporating water drop located in a superheated steam flow is studied using singular perturbation methods.The perturbation scheme involves three regions and some results concerning the Nusselt number and the drag coefficient of the drop are presented.     

Particle and drop dynamics in the flow behind a shock wave

The results of an investigation of the dynamics of hard particles and liquid drops in the flow behind a transmitted shock wave are presented. From the equation of motion of a particle in the shock wave, relations for the displacement, velocity and acceleration as functions of time and certain velocity-relaxation parameters taking into account the properties of the gas and the aerodynamic drag of the particles are obtained for unsteady flow around the particles at an acceleration of 10³–10⁴ m/s². It is shown that the velocity-relaxation parameters are universal. Approaches to finding the aerodynamic drag of freely-accelerating bodies from the dynamics of their acceleration after being suddenly exposed to the flow are considered. It is established that under these conditions the drop dynamics observed can be well described in terms of the same velocity-relaxation parameters with account for linear growth of the transverse drop size. All the kinematic functions obtained are confirmed experimentally.     

Two-phase two-component interface drag coefficients in separated phase flows

This work examines the behavior of the interface friction factor or drag coefficient as a means for extending the modeling of separated two-phase flows through the separate consideration of each phase. The model development of this work builds primarily upon the work of Carofano & McManus (1969), Wallis (1970) and Smith (1968). A one-dimensional flow model was developed for the case of vertical upward annular fiow of an air-water mixture with droplet entrainment. The model was developed for the investigation of accelerating flows in a nozzle but is utilized in this study for the investigation of momentum transport occurring in non-accelerating flows. This study presents experimental data showing the behaviour of the flow pressure drop occurring at various flow qualities and gas velocities. Also presented are empirical results for values of the air-water interface drag coefficient as a function of flow quality and gas core Reynolds number. The drag coefficient variation is compared to a previous correlation developed by Wallis 1969).     

A Taylor drop rising in a liquid co-current flow

The present work presents a numerical study on the behavior of isolated liquid Taylor drops rising in vertical tubes with co-current heavier continuous phase. Numerical simulations were performed with a previously validated model, implementing Volume of Fluid method in an axisymmetric geometry. Detailed flow patterns and drop shapes are provided and discussed for several conditions. The balance between gravity effect and velocity of the continuous phase flow was found to have a great influence in the flow patterns observed. The increase of inertial effects, due to the increase of Eo number and the co-current velocity, leads to the occurrence of closed recirculations below the drops. Furthermore, the continuous phase stabilization distance below the drop is a function of the drop Reynolds number. Drop and continuous phase velocities relationship was studied. A viscosity ratio related term was appended to a pre-existing correlation. The flow in the absence of gravity was also studied. Results demonstrate that micro-scale flow is a lower limit to the cases studied in the present work and suggest that the viscosity ratio affects mainly the inertial part of the drop velocity.     

具有弱夹层的封闭油藏中的瞬态渗流

多层油藏分为层间无窜流的油藏和层间有窜流的油藏。本文研究了层间有弱渗透夹层的两层油藏中两种情形的瞬态渗流,包括单层打开的瞬态渗流和一层注入、一层采出的两层油藏的瞬态渗流。用面积平均方法求出了弱渗透夹层封闭油藏中瞬态渗流的平均压降。分析表明:在渗流前期,层间窜流量逐渐增大,在后期层间窜流量趋于稳定;在单层开采的情况下:初期两层压降不同,非打开层的压降有滞后现象,并且隔层渗透率越小,油井工作强度越弱,滞后时间越长;后期两层压降趋于一致,可以作为一层油藏处理。在一注一采的油藏中,当注采量相同时,两层压降在后期趋于不同的稳定值;当注采量不同时,两层压降在后期趋于一致,但不能达到稳定。     

Diffusional interaction of drops in a liquid

The method of combining asymptotic expansions (with respect to a large Peclet number) is used to investigate the three-dimensional problem of steady-state convective diffusion to the surface of drops, around which flows a laminar stream of a viscous incompressible liquid whose velocity field is assumed to be known from the solution of the corresponding hydrodynamic problem. It is shown that for large Peclet numbers the heat and mass transfer between drops is completely determined by the mutual arrangement of special (starting or ending at the surface of a drop) lines of flow; under these circumstances, in the flow there are chains of drops which have no mutual diffusional effect on one another, and the total diffusional flow to a drop is determined by diffusion to particles located upstream in the same chain. For the case where the distance between the drops in the chain is much leas than P^1/2 (P is the Peclet number), formulas for the distribution of the concentration and the total diffusional flow to the surface of each drop are obtained. It is shown that the total diffusional flow to the surface of a drop approaches zero in inverse proportion to its order number in a chain, which generalizes [1], in which the axisymmetric case is considered. A solution of the diffusional case is obtained for the case where there are critical lines at the surface of the drop. The problem is solved to the end if the singular flow lines are not closed and depart to infinity. With the presence of a region of closed circulation behind the drops, the problem is reduced to an integral equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika, Zhidkosti i Gaza, No. 2, pp. 44–56, March–April, 1978.The author thanks Yu. P. Gupalo and Yu. S. Ryazantsev for their interest in the work.     

Drag coefficient of an absorbing plate set transverse to a flow

Supersonic flow past an absorbing surface is considered. The flow past a plate set transverse to the oncoming stream is calculated on the basis of a model kinetic equation. It is found that the dependence of the drag coefficient on the surface adsorption coefficient qualitatively changes on transition from the free molecular to the continuum flow regime. It is shown that in dense media a drag coefficient maximum is reached at total gas absorption by the surface. Under these conditions the surface interacts with the undisturbed stream. The effect of the extent of the plate-produced disturbance region on the plate drag is investigated.     

Effect of distributed gas injection on aerodynamic characteristics of a body of revolution in a supersonic flow

Results of experimental and numerical investigations of the effect of gas injection through a permeable porous surface on the drag coefficient of a cone-cylinder body of revolution in a supersonic flow with the Mach number range M _h = 3–6 are presented. It is demonstrated that gas injection through a porous nose cone with gas flow rates being 6–8% of the free-stream flow rate in the mid-section leads to a decrease in the drag coefficient approximately by 5–7%. The contributions of the decrease in the drag force acting on the model forebody and of the increase in the base pressure to the total drag reduction are approximately identical. Gas injection through a porous base surface with the flow rate approximately equal to 1% leads to a threefold increase in the base pressure and to a decrease in the drag coefficient. Gas injection through a porous base surface with the flow rate approximately equal to 5% gives rise to a supersonic flow zone in the base region.