A NUMERICAL SOLUTION OF THE FLOW AROUND A SPHERE IN A CIRCULAR CYLINDER

A numerical solution of the Navier-Stokes equations is presented for Poiseuille flow around an axially placed, fixed sphere in a circular cylinder. Streamlines and isovorticity lines are calculated from the governing equations for the strearnfunction and the vorticity. Isobars are calculated from a Poisson equation, derived from the Navier-Stokes equations. The pressure and vorticity distribution on the surface of the sphere, the additional pressure drop and the drag coefficients are presented. Solutions are obtained for Reynolds numbers up to 150 (based on cylinder diameter and mean velocity). The wall effects are examined by comparison with results of previous investigations for an unbounded flow around a sphere.     

THE TRANSIENT MOTION OF A SPHERICAL FLUID DROPLET

A numerical method is developed for investigation of the unsteady motion of a spherical fluid droplet under the influence of gravity. This study extends previous work valid for creeping flow to moderate Reynolds number. The unsteady flow fields inside and outside of the fluid sphere are described by the two-dimensional, axisymmetric Navier-Stokes equations in the form of vorticity and stream function, along with the equation of motion of the droplet. The governing equations are approximated by a central difference and a second-order upwind difference, and are solved iteratively using the Gauss-Siedel and secant methods. Numerical results of the time-dependent vorticity, stream function and drop velocity are presented for a water droplet moving through air and for an air bubble rising in water. The steady state drop velocity and the drag coefficient at various Reynolds numbers are examined, and they are shown to agree very well with previous results.     

Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—Part 1: Poiseuille flow (numerical)

At very low Reynolds numbers, we calculate the drag force exerted on a circular cylinder in cross flow fixed midway between two parallel plane walls which are fixed while the fluid experiences a Poiseuille profile at upstream and downstream. The drag wall correction factor is numerically investigated from a very weak interaction to the lubrication regime. The Navier-Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear co-ordinates. These equations are solved with using a finite differences method. The generation of the grid was carried out by the singularities method. We calculated the separate contributions of the pressure and viscous forces numerically. At very weak interactions, our numerical results are in good agreement with those obtained analytically by Harrison (Trans. Camb. Phil. Soc. 23 (1924) 71) and Faxèn (Proc. Roy. Swed. Acad. Eng. Sci. 187 (1946) 1). In the lubrication regime these numerical calculations are in very good agreement with those we carried out by asymptotic expansion. So that, the accuracy of the numerical code is tested. This analysis allowed us to show how that the pressure term prevails over the viscosity term in the lubrication regime. At very weak interaction, these forces have the same value.     

Numerical Simulation of the Lock-In Effect of a Fixed-Fixed Tube in Cross Flow

This paper deals with the development of a numerical calculation code that is able to simulate the three-dimensional flow through a heat exchanger tube bundle and therefore allows a coupled calculation of fluid-structure interaction between the flow and the tube bundle. The incompressible flow field is calculated by a Navier-Stokes solver using a first-order power law scheme, a SIMPLEC algorithm to calculate the pressure and velocity correction fields, and a line-by-line Gauss-Seidl tridiagonal algorithm to solve the linearised system of equations. The transient parts of the Navier-Stokes equations are discretised by a second-order forward finite differencing scheme. The turbulence is examined with the aid of a large-eddy turbulence model. The transient fluid forces acting on the tubes are calculated by integration of all local flow pressure values on the surfaces of the tubes. As an example a single fixed-fixed cylinder in a flow channel is considered using the structural calculation part already developed as well as the new flow field and flow forces subroutines. The time series of the tube's motion and the fluid forces acting on the tube are analysed by Fourier's transformation. The lock-in effect occurring when the vortex shedding frequency approaches the first natural frequency of the tube can be excellently demonstrated by varying the inflow velocity over a wide range of Reynolds numbers.     

Modeling of bubble-column flows with quadrature-based moment methods

Bubble-column reactors are frequently employed in the biological, chemical and petrochemical industries. This paper presents a novel approach to model bubble-column flows using quadrature-based moment methods (QBMM). A fully two-way coupled flow solver is developed that solves the incompressible Navier-Stokes equation for the liquid phase and moment transport equations for the dispersed bubble phase. The moment transport equations for the dispersed bubble phase are solved using a kinetic theory approach. Contributions from the liquid-phase pressure gradient, vorticity, drag, virtual mass and gravity are accounted for in the bubble-phase force balance. The solution algorithm and coupling procedure are described in detail, and results are presented for a 2-D bubble column with two different gas flow rates (1.6 and 8.0 l/min).     

Steady incompressible fluid flow over a bundle of cylinders at moderate reynolds numbers

The complete Navier-Stokes equations describing the steady flow of incompressible Newtonian fluids normal to an array of long cylinders have been solved numerically using the finite difference method in terms of the stream function and vorticity variables. The inter-cylinder interactions have been mimicked using the well known free surface cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, stagnation pressure, stream line and iso-vorticity line plots, as well as on the values of the integral quantities, such as pressure, friction and total drag coefficients, have been obtained under wide ranges of conditions as follows: 0.3 ≤ ? 0.99 and 0.01 ≤ Re ≤ 100. The numerical results presented herein have been validated using the appropriate theoretical and experimental results available in the literature; the match between the present predictions and the scant experimental results is good.     

Low Reynolds number flow of power-law fluids over two square cylinders in tandem

The governing partial differential equations have been solved numerically for the 2-D and steady powerlaw fluid flow over two square cylinders in tandem arrangement. Extensive numerical results spanning wide ranges of the governing parameters as Reynolds number (0.1≤Re≤40), power-law index (0.2≤n≤1) and inter-cylinder spacing (2≤L/d≤6) are presented herein; limited results for L/d=24 are also obtained to approach the single cylinder behavior. The detailed flow visualization is done by means of the streamline and vorticity contours in the vicinity of two cylinders. The global characteristics are analyzed in terms of the surface pressure distribution and pressure drag coefficient. The drag coefficient shows the classical inverse dependence on the Reynolds number irrespective of the value of the powerlaw index; the drag on the upstream cylinder is always greater than that for the downstream cylinder.     

Flow of non‐Newtonian polymeric solutions through fibrous media

The equations of motion (continuity and momentum) describing the steady flow of incompressible power law liquids in a model porous medium consisting of an assemblage of long cylinders have been solved numerically using the finite difference method. The field equations as well as the pertinent boundary conditions have been re‐cast in terms of the stream function and vorticity. The inter‐cylinder interactions have been simulated using a simple “concentric cylinders” cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, streamlines, and viscosity distribution on the surface of the solid cylinder as well as on the values of the pressure and friction drag coefficients under wide ranges of physical (0.4 ≤ ϵ ≤ 0.95; 1 ≥ n ≥ 0.4) and kinematic (0.01 ≤ Re ≤ 10) conditions have been obtained. The numerical results presented herein have been validated using the experimental results for the flow of Newtonian and power law fluids available in the literature; the match between the present predictions and the experiments was found to be satisfactory. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 76: 1171–1185, 2000     

Impaction of spherical particles on cylinders at moderate Reynolds numbers

Inertial and interceptive impaction of spherical particles on circular cylinders was investigated theoretically. The particles were considered to be suspended in a fluid moving steadily through a random array of parallel cylinders. Fluid flow fields around the cylinders were obtained by numerically solving the Navier-Stokes Equations subject to Kuwabara's zero vorticity boundary condition. These solutions were subsequently utilized in calculating particle trajectories and impaction efficiencies. The latter are presented as functions of Reynolds number (0·2 ? Re_c ? 40), particle inertial parameter (0 ? P ? 1000), particle to cylinder size ratio (0·001 ? K ? 1·0) and cylinder concentration (10^?4 ? c ? 0·111).The impaction efficiencies and critical inertial parameters differ from earlier theoretical predictions. The discrepancies are primarily attributable to the inaccurate flow field representations used by previous authors. The agreement between Subramanyam and Kuloor's experimental work and present theory is satisfactory.     

Transient three-dimensional heat transfer from rotating spheres with surface blowing

Transient heat transfer and thermal patterns around a rotating spherical particle with surface blowing are studied numerically for Reynolds numbers in the range 10?Re?300 and non-dimensional angular velocities up to Ω=1. This range of Reynolds number includes three distinct wake regimes: steady and axisymmetrical, steady but non-symmetrical, and unsteady with vortex shedding. The Navier-Stokes and energy equations for an incompressible viscous flow are solved numerically by a finite-volume method in a three-dimensional and time-accurate manner. The transient aspects of the thermal wakes associated with the aforementioned wake regimes have been explored. An interesting feature associated with particle rotation and surface blowing is that they can affect the near wake structure in such a way that an unsteady three-dimensional flow with vortex shedding develops at lower Reynolds numbers as compared to flow over a solid sphere in the absence of these effects, and thus, the temperature distributions around the particle are significantly affected. Despite the fact that particle rotation brings about major changes locally, the surface-averaged heat transfer rates are not influenced appreciably even at high rotational speeds; consequently, it is shown that the total heat transfer rates associated with rotating spheres with surface blowing can be calculated from heat transfer correlations developed for flow over evaporating droplets.     

Effect of bubble size in turbulent bubbly downflow in a vertical channel

The effect of bubble size on the properties of downward turbulent flows of bubbly liquids in a vertical channel is examined using direct numerical simulations, where the full Navier-Stokes equations are solved by a parallelized front-tracking/finite-volume method. The turbulent channel flow is driven downward by an imposed constant pressure gradient, and the friction Reynolds number, based on the friction velocity and half-width of the channel, is 127.3. Bubbles of two different sizes, with diameters of 31.8 and 38.2 wall units, are introduced into the turbulent flow with a monodisperse or bidisperse distribution. The results show that for the cases studied here the bubble size has little effect on the void fraction distribution and the mean vertical velocity profile. The average velocity fluctuations and the vorticity profiles across the channel do, however, change.     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

Flow of viscous fluid through a circular aperture

The creeping flow of a highly viscous incompressible fluid through a circular aperture located in an infinitely wide horizontal plate is analyzed by solving Navier-Stokes equations without inertia terms. Solutions for vertical and radial velocities as well as pressure have been obtained in terms of integral equations with an undetermined Kernal function. This function has been evaluated by assuming several different velocity distributions at the aperture, and the corresponding pressure drop for each case has been calculated. The results show that the pressure loss for a given flow rate goes through a minimum as the assumed velocity profile changes from flat to parabolic. Based on the minimum energy dissipation theorem of Helmholtz, the most appropriate velocity distribution is discussed. Experimental data obtained using sharp-edged orifices are compared with theoretical predictions.     

Study on the floatation mechanism and floatation performance for the floatation cyclone of natural inlet air

A novel structure floatation cyclone of natural inlet air has been designed, and its structural characteristics and floatation principle have been analyzed. The velocity and pressure distributions within the flow field of the floatation cyclone have been studied by Navier-Stokes equations. Based on the flow characteristics of the mixture of fine coal and water, reasonable boundary conditions are decided and the equations are modified, so that the final equations can describe the real flow state of the flow field of the floatation cyclone. The boundary surface position between float coal and tailings is determined and there is an air cylinder in the central region of the floatation cyclone. The research reveals the floatation mechanism of the floatation cyclone. The floatation results can be greatly improved by regulating the structural dimensions of cyclone. The experimental results show that the floatation cyclone is very effective for the floatation of fine coal grains.     

Steady flow of power-law fluids across an unconfined elliptical cylinder

The momentum transfer characteristics of the power-law fluid flow past an unconfined elliptic cylinder is investigated numerically by solving continuity and momentum equations using FLUENT (version 6.2) in the two-dimensional steady cross-flow regime. The influence of the power-law index (0.2?n?1.8), Reynolds number (0.01?Re?40) and the aspect ratio of the elliptic cylinder (0.2?E?5) on the local and global flow characteristics has been studied. In addition, flow patterns showing streamline and vorticity profiles, and the pressure distribution on the surface of the cylinder have also been presented to provide further physical insights into the detailed flow kinematics. For shear-thinning (n<1) behaviour and the aspect ratio E>1, flow separation is somewhat delayed and the resulting wake is also shorter; on the other hand, for shear-thickening (n>1) fluid behaviour and for E<1, the opposite behaviour is obtained. The pressure coefficient and drag coefficient show a complex dependence on the Reynolds number and power-law index. The decrease in the degree of shear-thinning behaviour increases the drag coefficient, especially at low Reynolds numbers. While the aspect ratio of the cylinder exerts significant influence on the detailed flow characteristics, the total drag coefficient is only weakly dependent on the aspect ratio in shear-thickening fluids. The effect of the flow behaviour index, however, diminishes gradually with the increasing Reynolds number. The numerical results have also been presented in terms of closure relations for easy use in a new application.     

Mathematical model for gas flow through a packed bed in the presence of sources and sinks

Flow within a packed bed is normally calculated by attempting to simultaneously satisfy the continuity and Ergun equations. However, the presence of gas sources/sinks within the bed escalates the complexity of the problem, particularly when the flow is two-dimensional and a solution to the full Ergun equation is required. In quest of an efficient and dependable algorithm for the calculation of gas flow, a critical review of existing solution methods was undertaken and a new method, ‘FLOW’, is now proposed. The technique retains the viscous and inertial pressure gradient terms of the Ergun equation, and both are treated as linear functions of the flow. Solutions are approached iteratively; using finite difference techniques, the continuity and linearized Ergun equations are solved for the pressure field; a new flow field is then calculated from which is derived an adjustment to the inertial resistance term of the Ergun equation. The sequence is repeated until satisfactory convergence is obtained. Relatively few iterations are normally required and, for the case of negligible inertial pressure drop, one calculation cycle is sufficient. A comparison of results obtained using the ‘FLOW’, modified ‘SIMPLE’ and vorticity procedures is presented. The proposed method allow flexibility in the specification of boundary conditions and can be applied to compressible or incompressible flow, as well as for the case of nonisothermal beds.     

START-UP FLOW IN A PIPE FOLLOWING THE SUDDEN IMPOSITION OF A CONSTANT FLOW RATE

The transient flow in a long pipe following the sudden imposition of a constant flow rate is considered. An exact analytical solution of the axisymmetric Navier-Stokes equations in terms of an infinite series of Bessel functions is derived. Time-dependent velocity profiles, as well as time histories of the axial pressure-gradient and the wall-friction, are presented. It is observed that the start-up time required to reach steady state is significantly shorter than if the start-up flow results from a suddenly imposed constant pressure gradient. The “Annular Jet Effect” observed experimentally by Kataoka et al. (1975) is not exhibited by the present solution.     

NATURAL CONVECTION INSIDE A HORIZONTAL CYLINDER

Natural convection of a fluid contained in an infinitely long horizontal cylinder at large Prandtl number and unit-order Grashof number is analyzed. The motion is generated by an imposed cosine wall temperature distribution which includes an arbitrary phase angle. The phase angle is a measure of the location of the wall temperature extrema.

From an asymptotic ordering of the energy and vorticity transport equations for large Prandtl number it is shown that the core region, which contains fluid surrounded completely by a boundary-layer flow along the cylinder wall, may assume either of two configurations.

For heating angles near the heating-from-the-side case (wall temperature extrema at the ends of the horizontal diameter) linearized forms of the boundary-layer equations are developed which yield solutions that match the core configuration not considered previously. The form of the results agrees generally with experimental evidence for heating-from-the-side.     

NATURAL CONVECTION INSIDE A HORIZONTAL CYLINDER

Natural convection of a fluid contained in an infinitely long horizontal cylinder at large Prandtl number and unit-order Grashof number is analyzed. The motion is generated by an imposed cosine wall temperature distribution which includes an arbitrary phase angle. The phase angle is a measure of the location of the wall temperature extrema.

From an asymptotic ordering of the energy and vorticity transport equations for large Prandtl number it is shown that the core region, which contains fluid surrounded completely by a boundary-layer flow along the cylinder wall, may assume either of two configurations.

For heating angles near the heating-from-the-side case (wall temperature extrema at the ends of the horizontal diameter) linearized forms of the boundary-layer equations are developed which yield solutions that match the core configuration not considered previously. The form of the results agrees generally with experimental evidence for heating-from-the-side.     

高粘度流体二维搅拌流场

用有限元法求解二维搅拌流动的Navier-Stokes方程。采用极坐标系,引用罚函数法处理流体不可压缩条件,并消去压力项,成功地计算了锚式桨搅拌槽二维流场的速度分布、流型以及剪切率分布。计算的流型与实验拍摄的结果符合良好,还对不同Re数下的搅拌流场进行了数值模拟,并描述了典型Re数下的计算结果及Re数对流场的影响。