Drag of a rotating sphere

We consider the problem of steady incompressible viscous fluid flow about a rotating sphere, with the flow specified on a sphere of finite radius, which reduces to the solution of the complete Navier-Stokes equations.The dimensionless stream functions and circulai velocity are sought in the form of series in powers of the Reynolds numbers, which converge for small values of this number. Recurrence formulas are derived for determining the coefficients of these series. The pressure, rotational resistance torque, and drag are determined. It is established that the rotating sphere has higher drag than a stationary sphere. The leading term of the series in powers of the Reynolds number for the drag and resistive torque is calculated.     

Cessation of the lid-driven cavity flow of Newtonian and Bingham fluids

We provide benchmark results for a transient variant of the lid-driven cavity problem, where the lid motion is suddenly stopped and the flow is left to decay under the action of viscosity. Results include Newtonian as well as Bingham flows, the latter having finite cessation times, for Reynolds numbers Re ∈ [1, 1000] and Bingham numbers Bn ∈ [0, 10]. The finite-volume method and Papanastasiou regularisation were employed. A combination of Re and Bn, the effective Reynolds number, is shown to convey more information about the flow than either Re or Bn alone. A time scale which characterises the flow independently of the geometry and flow parameters is proposed.     

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.     

Creeping-flow rotation of a slip spheroid about its axis of revolution

The problem of rotation of a rigid spheroidal particle about its axis of revolution in a viscous fluid is studied analytically and numerically in the steady limit of negligible Reynolds number. The fluid is allowed to slip at the surface of the particle. The general solution for the fluid velocity in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of separation of variables. The slip boundary condition on the surface of the rotating particle is applied to this general solution to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary collocation method or explicit formulas derived analytically. The torque exerted on the spheroidal particle by the fluid is evaluated for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic torque results and the available analytical solutions in the limiting cases is good. It is found that the torque exerted on the rotating spheroid normalized by that on a sphere with radius equal to the equatorial radius of the spheroid increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip or finite-slip spheroid and vanishes for a perfectly slip spheroid. For a spheroid with a specified aspect ratio, the torque is a monotonically decreasing function of the slip capability of the particle.     

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     

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.     

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.     

Effect of fluid temperature and uniformly accelerated rotation of the inner sphere on the selection of the flow pattern in a spherical layer

The process of the selection of one of the two flow patterns possible in the hysteresis region, when the Reynolds number is varied in different directions, and differing with respect to the azimuthal wavenumber, 3 or 4, is experimentally investigated. The flow pattern selection proceeds under the influence of an increase in the rotation velocity of the inner sphere at a constant acceleration, the post-acceleration velocity remaining constant. The spherical layer thickness is equal to the inner sphere radius and the outer boundary is fixed. It is established that there is a time lag between the beginning (end) of the sphere acceleration and the beginning (end) of the variation in the measured azimuthal velocity component. It is found that the acceleration necessary for one flow pattern to be replaced by the other significantly depends not only on the Reynolds numbers at which the acceleration begins and ends but also on the fluid temperature in the layer. It is shown that the temperature dependence can be attributed to the variation in the Reynolds number corresponding to the position of the hysteresis boundary when the working fluid viscosity is varied in the layer.     

Hydrodynamic force and torque models for a particle moving near a wall at finite particle Reynolds numbers

This research work is aimed at proposing models for the hydrodynamic force and torque experienced by a spherical particle moving near a solid wall in a viscous fluid at finite particle Reynolds numbers. Conventional lubrication theory was developed based on the theory of Stokes flow around the particle at vanishing particle Reynolds number. In order to account for the effects of finite particle Reynolds number on the models for hydrodynamic force and torque near a wall, we use four types of simple motions at different particle Reynolds numbers. Using the lattice Boltzmann method and considering the moving boundary conditions, we fully resolve the flow field near the particle and obtain the models for hydrodynamic force and torque as functions of particle Reynolds number and the dimensionless gap between the particle and the wall. The resolution is up to 50 grids per particle diameter. After comparing numerical results of the coefficients with conventional results based on Stokes flow, we propose new models for hydrodynamic force and torque at different particle Reynolds numbers. It is shown that the particle Reynolds number has a significant impact on the models for hydrodynamic force and torque. Furthermore, the models are validated against general motions of a particle and available modeling results from literature. The proposed models could be used as sub-grid scale models where the flows between particle and wall can not be fully resolved, or be used in Lagrangian simulations of particle-laden flows when particles are close to a wall instead of the currently used models for an isolated particle.     

Steady Ostwald flow between two differentially rotating co-axial discs

The steady flow of a power-law (Ostwald) fluid between two differentially rotating, parallel, co-axial discs has been considered for both large and small Reynolds number. All edge effects of the discs are neglected, the discs rotate in the same sense and the distance between the two discs is much smaller than their radius. In the large Reynolds number case a similarity solution is sought. It is assumed that the flow consists of boundary layers on the discs, while the core rotates as a rigid body with speed intermediate of those of the discs. The boundary layer is thinner than in the equivalent Newtonian problem, and the decay of the boundary layers is found to be algebraic. This slow decay contrasts with the faster exponential decay in the Newtonian case. For the low Reynolds number problem, the ratio of the disc separation to radius was taken to be much smaller than the Reynolds number. This is, in effect, a lubrication-type problem. The velocity components are expressed as expansions in ascending powers of the Reynolds number. For both the large and small Reynolds number flow, the torque is calculated as a function of the disc speeds, for various values of the power-law index n.     

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re ₁ , Re ₂ ) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states. Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by M.Y. Hussaini     

Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus

This paper presents results obtained by employing a modified Galerkin finite element method to analyse the steady state flow of a fluid contained between two concentric, rotating spheres. The spheres are assumed to be rigid and the cavity region between the spheres is filled with an incompressible, viscous, Newtonian fluid. The inner sphere is constrained to rotate about a vertical axis with a prescribed angular velocity, while the outer sphere is fixed. Results for the circumferential function Ω, streamfunction ψ, vorticity function ζ and inner boundary torque T₁ are presented for Reynolds numbers Re ? 2000 and radius ratios 0.1 ? α ? 0.9. The method proved effective for obtaining results for a wide range of radius ratios (0.1 ? α ? 0.9) and Reynolds numbers (0 ? Re ? 2000). Previous investigators who employed the finite difference method experienced difficulties in obtaining results for cases with radius ratios α ? 0.2, except for small Reynolds numbers (Re ? 100). Results for Ω, Ψ, ζ and T₁ obtained in this study for radius ratios 0.8 ≤ α ≤ 0.9 verified the development of Taylor vortices reported by other investigators. The research indicates that the method may be useful for analysing other non-linear fluid flow problems.     

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.     

Response of the Sphere Wake to Freestream Fluctuations

Direct numerical simulations have been used to investigate the response of the wake of a sphere to freestream fluctuations. This study has been motivated by the need to understand particle-induced turbulence enhancement in particulate flows. A sequence of simulations of flow past a sphere have been carried out where the frequency and amplitude of the freestream fluctuations and the flow Reynolds number has been varied systematically. It has been suggested that turbulence enhancement is primarily caused by vortex shedding from particles (Gore and Crowe, 1989; Hetsroni, 1989). Our simulations of the forced wake indicate that turbulence enhancement may be attributed to natural vortex shedding only when the freestream fluctuation level is low and the Reynolds number is greater than about 300. In addition to natural vortex shedding, the current simulations also suggest another mechanism for turbulence enhancement. It is found that in the presence of freestream fluctuations, the wake behaves like an oscillator and returns large amounts of kinetic energy to the surrounding fluid at resonance. This mechanism is not associated with natural vortex shedding and is therefore capable of enhancing freestream turbulence even at Reynolds numbers less than 300. Simulations also indicate that when the turbulence intensity of the carrier fluid is high, this resonance mechanism might be solely responsible for turbulence enhancement. Finally, our simulations also suggest a possible explanation for the correlation between turbulence enhancement and the ratio of the particle size to the size of energy containing eddies of turbulence found by Gore and Crowe (1989). Received 5 October 1999 and accepted 14 October 1999     

Second-order finite difference solutions for the flow between rotating concentric spheres

This paper describes a second-order method to calculate approximate solutions to flow of viscous incompressible fluid between rotating concentric spheres. The governing partial differential equations are presented in the stream–vorticity formulation and are written as a series of second-order equations. The technique employed makes use of second-order approximations for all terms in the governing equations and is dependent upon the direction of flow at a given point. This upwind technique has allowed us to generate approximate solutions with larger Reynolds numbers than has generally been possible for second and higher-order techniques. Solutions have been obtained with Reynolds numbers as large as 3000 and with grids as fine as a 40 × 40 mesh. Results are displayed in the form of level curves for both the stream and vorticity functions. A dimensionless quantity related to the torque acting on both spheres has been calculated from the approximate solution and compared with other results. Results with smaller Reynolds numbers such as 100 and 1000 are in excellent agreement with other published results.     

Stokes flow of a rotating sphere around the axis of a circular orifice or a circular disk

This paper presents an infinite series solution to the creeping flow equations for the axisymmetric motion of a sphere of arbitrary size rotating in a quiescent fluid around the axis of a circular orifice or a circular disk whose diameters are either larger or smaller than that of the sphere. Numerical tests of the convergence are passed and the comparison with the exact solution and other computational results shows an agreement to five significant figures for the torque coefficients in both cases. The torque coefficients are obtained for the sphere located up to a position tangent to the wall plane containing either the orifice or the disk. It is concluded that the torque coefficients of the sphere and the disk are monotonically increasing with the decrease of the distance from the disk or the orifice plane in both cases.     

Numerical simulation on steady flow around and through a porous sphere

The flow around and through a porous sphere is investigated numerically. It is found that the recirculating wake existing downstream of the porous sphere is either completely detached from or penetrates it. Specially, in a certain range of Darcy number, the wake may initially increase in size with an increase in Reynolds number but then decrease in size and eventually disappear when the Reynolds number is sufficiently large. For a small Darcy number, the critical Reynolds number for the onset of recirculating wake may be significantly smaller than that of a solid sphere. The present study suggests that the surface curvature has an important effect on the initial position of the onset of recirculating wake.     

The electrostatic torque on a rotating conducting sphere

The electrostatic torque on a rotating sphere through which a current is fed by means of two diametrically situated sliding contacts is positive in the direction of rotation. However, this torque is too small in practice to keep a ball bearing motor running.     

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.     

The electromagnetic torque on axially symmetric rotating metal cylinders and spheres

The torque is calculated on an electrically resistive rotating cylinder and sphere through which a current is fed by means of sliding contacts. The torque on a rotating cylinder is proven to vanish identically for arbitrary angular velocity. The torque on the sphere is shown to vanish up to second order in an expansion with the angular velocity as the expansion parameter. The nonzero torque in first order found by Gruenberg is shown to be due to an algebraic error.The author wrote this article in honour of Bert Broer, who he considers an exponent of classical physics viewed from a modern and general point of view. Bert would undoubtedly have written on the present subject in a broader sense.