Flow of a viscous fluid around a sphere of finite radius

We consider the stead y flow of a viscous fluid around a sphere of finiteradius in non-linear formulation. The equations of motion are written in non-dimensional terms. We seek their solution as the expansion of the unknown stream function in a series of powers of the Reynolds number, the coefficients of which are polynomials in associated Legendre functions of the first kind. Recurrence relations are given for the sequential determination of all coefficients. The velocity and pressure fields are determined. The drag is calculated. Numerical calculations are carried out.     

Lift force on rotating sphere at low Reynolds numbers and high rotational speeds

The lift force on an isolated rotating sphere in a uniform flow was investigated by means of a three-dimensional numerical simulation for low Reynolds numbers (based on the sphere diameter) (Re<68.4) and high dimensionless rotational speeds (Г5). The Navier-Stokes equations in Cartesian coordinate system were solved using a finite volume formulation based on SIMPLE procedure. The accuracy of the numerical simulation was tested through a comparison with available theoretical, numerical and experimental results at low Reynolds numbers, and it was found that they were in close agreement under the above mentioned ranges of the Reynolds number and rotational speed. From a detailed computation of the flow field around a rotational sphere in extended ranges of the Reynolds number and rotational speed, the results show that, with increasing the rotational speed or decreasing the Reynolds number, the lift coefficient increases. An empirical equation more accurate than those obtained by previous studies was obtained to describe both effects of the rotational speed and Reynolds number on the lift force on a sphere. It was found in calcttlations that the drag coefficient is not significantly affected by the rotation of the sphere. The ratio of the lift force to the drag force, both of which act on a sphere in a uniform flow at the same time, was investigated. For a small spherical particle such as one of about 100μm in diameter, even if the rotational speed reaches about 10^6 revolutions per minute, the lift force can be neglected as compared with the drag force.     

球栅的Oseen流动及阻力的尾流效应

本文通过大小相同且等距放置的球栅的轴对称Oseen流动研究了阻力的尾流效应。在每个球的球心放置Oseen流子得到了问题的级数解。截断无穷级数并采用配置法解线性代数方程组求出了球栅Oseen流动的近似解及每个球所遭受的阻力。 在不同的球的个数,不同的球的间距以及不同的雷诺数下计算了各个圆球的阻力系数,发现除球栅的遮蔽效应和端缘效应外还存在着尾流效应。研究了上述参数对这些效应的影响并与Stokes流动的结果进行了比较。文章还对方法的收敛性进行了数值研究。     

The oseen flow of finite clusters of spheres and the wake effect on the drag factor

In this paper, the wake effect on drag factor in the axisymmetric Oseen flow of the finite clusters of equally spaced spheres with same size is studied. Putting the Oseen lets on the centres of all the spheres, the series solution of the problem is obtained. By truncating the infinite series and applying the collocation method to solve a set of the linear algebraic equations, the approximate solution of the Oseen flow of finite clusters of spheres and the drag factor for each sphere are presented. The effect of the sphere number and spacing on the drag factor of each sphere under different Reynolds numbers are calculated and the wake effect as well as the shielding effect and the end effect are revealed. The influence of various parameters on the effects is considered and compared with the corresponding results of the Stokes flow. The convergence of the method is also studied numerically in this paper.     

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

Study on aerodynamic force acting on a sphere with and without boundary layer trips around the critical Reynolds number with a magnetic suspension and balance system

Aerodynamic force acting on a sphere for five kinds of boundary layer trips around the critical Reynolds number, together with the force on a smooth sphere, was successfully measured. This was achieved using JAXA’s 60-cm Magnetic Suspension and Balance System after performing detailed simulations and adjusting the sphere mass and its control parameters. The minimum drag coefficient of a smooth sphere was evaluated around 0.19 in the support-interference-free condition. No hysteresis was observed for the drag coefficient in the critical range for tested sphere with boundary layer trips. Using three serially connected 2nd-order Butterworth low-pass filters, an inertia force oscillating at less than 15 Hz was evaluated from the measured model position, and the unsteady aerodynamic force acting on the sphere was also evaluated with reasonable accuracy. Two kinds of oscillatory aerodynamic forces appeared in the critical range depending on the sphere surface condition: a force rotating around an axis parallel to the uniform flow for both a smooth sphere and a sphere with axially symmetric 0.17-mm-high backward step, and an oscillating force in the plane including the axis parallel to the flow for a sphere with axially symmetric step implemented with 0.35–mm-thick tape with wrinkles acting as small vortex generators. There was also observed a force irregularly rotating through less than 180° in the range about a sphere axis parallel to the flow for a smooth sphere in the supercritical range.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

Flow past a rotating sphere in a Bingham plastic fluid,up to a Reynolds number of 10,000

The flow induced by a sphere rotating inside a non-Newtonian, Bingham fluid has been investigated numerically. The rotating sphere is enclosed in a concentric cubic box with solid boundaries. The Bingham number varied between 0.01 and 100,000 and the Reynolds number varied between 0.01 and 10,000. The torque increases with the Bingham number and reaches an asymptotic state at large Bn. The torque is independent of the Reynolds number at high Bn. The yielded region around the sphere has been determined and an empirical equation is found for its extent.     

Isothermal flow past a blowing sphere

Steady, axisymmetric, isothermal, incompressible flow past a sphere with uniform blowing out of the surface is investigated for Reynolds numbers in the range 1 to 100 and surface velocities up to 10 times the free stream value. A stream-function-velocity formulation of the flow equations in spherical polar co-ordinates is used and the equations are solved by a Galerkin finite-element method. Reductions in the drag coefficients arising from blowing are computed and the effects on the viscous and pressure contributions to the drag considered. Changes in the surface pressure, surface vorticity and flow patterns for two values of the Reynolds number (1 and 40) are examined in greater detail. Particular attention is paid to the perturbation to the flow field far from the sphere.     

Flow around a sphere with material cross flow at low reynolds numbers

An analytical solution is carried out for the problem of the flow around a sphere with material cross flow at Reynolds numbers less than 1 and a blowing velocity less than the free stream velocity. The method of asymptotic expansions of Pearson and Proudman is used for the solution. Expressions are obtained for the distribution of the current and velocity component functions as well as for the aerodynamic drag coefficient of the sphere. It is shown that blowing diminishes the sphere drag, where its influence will increase as the Reynolds number grows.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 103–109, May–June, 1972.     

SERIES PERTURBATIONS APPROXIMATE SOLUTIONS TO N-S EQUATIONS AND MODIFICATION TO ASYMPTOTIC EXPANSION MATCHED METHOD

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Hydrodynamic forces acting on a rigid fixed sphere in early transitional regimes

A spectral – spectral-element code is used to investigate the hydrodynamic forces acting on a fixed sphere placed in a uniform flow in the Reynolds number interval [10–320] covering the early stages of transition, i.e. the steady axisymmetric regime with detached flow, the steady non-axisymmetric and the unsteady periodic regimes of the sphere wake. The mentioned changes of regimes, shown by several authors to be related to a regular and a Hopf bifurcations in the wake, result in significant changes of hydrodynamic action of the flow on the sphere. In the present paper, we show that the loss of axisymmetry is accompanied not only by an onset of lift but also of a torque and we give accurate values of drag, lift and torque in the whole interval of investigated Reynolds numbers. Among other results show, moreover, that each bifurcation is accompanied also by a change of the trend of the drag versus Reynolds number dependence, the overall qualitative effect of instabilities being an increase of drag.     

Numerical solution of viscous flows using integral equation methods

A formulation of the boundary element method for the solution of non-zero Reynolds number incompressible flows in which the non-linear terms are lumped together to form a forcing function is presented. Solutions can be obtained at low to moderate Reynolds numbers. The method was tested using the flow of a fluid in a two-dimensional converging channel (Hamel flow) for which an exact solution is available. An axisymmetric formulation is demonstrated by examining the drag experienced by a sphere held stationary in uniform flow. Performance of the method was satisfactory. New results for an axisymmetric free jet at zero Reynolds number obtained using the boundary element method are also included. The method is ideal for this type of free-surface problem.     

Laminar forced convection about rotating bodies at low Pr-number

Nusselt number is presented as a function of Reynolds number and Prandtl number (Pr) for a laminar flow about a non- rotating and rotating sphere. The sphere rotation is expressed in terms of Taylor number. Pr is ranging between 0.7 and 0.2. The governing equations were solved numerically using finite difference method for constant wall temperature and constant heat flux conditions. Received on 17 October 1996     

Hyperbolic Approximation of the Navier-Stokes Equations for Viscous Mixed Flows

Simplified two-dimensional Navier-Stokes equations of the hyperbolic type are derived for viscous mixed (with transition through the sonic velocity) internal and external flows as a result of a special splitting of the pressure gradient in the predominant flow direction into hyperbolic and elliptic components. The application of these equations is illustrated with reference to the calculation of Laval nozzle flows and the problem of supersonic flow past blunt bodies. The hyperbolic approximation obtained adequately describes the interaction between the stream and surfaces for internal and external flows and can be used over a wide Mach number range at moderate and high Reynolds numbers. Examples of the calculation of viscous mixed flows in a Laval nozzle with large longitudinal throat curvature and in a shock layer in the neighborhood of a sphere and a large-aspect-ratio hemisphere-cylinder are given. The problem of determining the drag coefficient of cold and hot spheres is solved in a new formulation for supersonic air flow over a wide range of Reynolds numbers. In the case of low and moderate Reynolds numbers a drag reduction effect is detected when the surface of the sphere is cooled.     

An experimental investigation of the flow about a sphere rotating in a rivlin-ericksen fluid

Tangential and radial velocity profiles were measured for the flow about a sphere rotating slowly in a Newtonian fluid, contained in a rectangular tank. Velocities were determined from enlarged streak photographs of aluminium particles moving in a collimated “sheet” of light, at several planes through the flow field. Similar velocity profiles were measured for the flow of a 1.50% Natrosol 250 H solution about two spheres of different diameters rotating in two different sized rectangular tanks. A set of velocity distributions were also measured for a sphere rotating in a 0.9% Natrosol 250 H solution. A dye tracer study of the flow about a sphere rotating in this liquid is presented as well. Both Natrosol solutions exhibited viscoelastic behaviour. The Newtonian fluid study was carried out at a Reynolds number of 1.2 and the viscoelastic fluid studies were within the Reynolds number range of 0.05–1.24.The zero shear viscosities of the Natrosol solutions were measured using the falling-sphere method. The non-Newtonian material parameters were obtained by fitting the theoretical curves to the measured velocity data. The values of the elastic and shear thinning parameters for the two liquids obtained in the different geometrical and dynamical situations are compared.     

Drag forces of interacting spheres in power-law fluids

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6 to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.     

Finite element simulations of incompressible flow past a heated/cooled sphere

This paper reports numerical simulation of the flow past a heated/cooled sphere. A Galerkin finite element method is used to solve the 3D incompressible Boussinesq equations in primitive variable form. Numerical simulations of flow around the sphere for a range of Grashof numbers and moderate Reynolds numbers, were conducted. The drag coefficient for adiabatic flow shows good agreement with standard correlations over the range of the Reynolds numbers investigated. It is shown that the drag can vary considerably with heating of the sphere and that computational fluid dynamics methods can be used to derive constitutive laws for macroscopic momentum and heat exchange in multiphase flow. © 1998 John Wiley & Sons, Ltd.     

Low Reynolds-number moment on asymmetric bodies

The moment induced on asymmetric particles by a low Reynolds-number flow is examined. A flat disk and a half sphere are suspended in low-velocity airflow and the moment exerted on the particles is measured as a function of Reynolds number and approach angle. Object Reynolds-numbers range between 20 and 200, and the measured moment is in the nNm range. A moment coefficient (a moment normalized with the drag force and object radius) is calculated from the moment measurements and found to generally decrease with increasing Reynolds number. Stable approach angle positions are identified for the objects in the flow, and it is shown that the objects tend to rotate towards the position of maximum drag (local extremum) from other approach angles. The present measurements are consistent with previous qualitative observations of free-falling particles.     

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.