Slow flow of gas around a strongly heated or cooled sphere

Problems similar to those considered in [1, 2] are studied, namely, slow flow over a uniformly heated (or cooled) spherical particle and flow past a weakly nonuniformly heated sphere in the absence of external body forces and with allowance for thermal stresses in the gas. The use of an improved method of numerical solution [3] has made it possible to advance into the region of large temperature differences. A new effect is found: allowance for the thermal stresses in the case of flow around a strongly heated sphere leads to the appearance of a suction force instead of a drag. In the case of flow around a nonuniformly heated sphere the influence of thermal stresses is unimportant. The problems are considered for two temperature dependences of the transport coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 170–175, October–December, 1981.     

The inlet effect on the drag factor of a sphere in a tube

The creeping motion Ground a sphere situated axisymmetrically near the entrance of asemi-infinite circular cylindrical tube is analyzed using infinite series solutions for thevelocity components. pressure and the stream function. Truncating the infinite series. thecorresponding coefficients in the series are determined by a collocation technique. The dragfactor and the stress distribution on the surface of the sphere are calculated for the sphere inmotion in quiescent fluid and for the flow with uniform velocity at the entrance past a rigidlyheld sphere. The results indicate that a sphere near the entrance which has a uniformentrance velocity profile will suffer larger drag than that in infinite tube.Theconvergence of the collocation technique is tested by numerical calculation. It is shown thatthe technique has good convergence properties.     

Flow of a viscous fluid past a heterogeneous porous sphere at low Reynolds numbers

A flow past a heterogeneous porous sphere is investigated by using the perturbation theory. The flow through the sphere is divided into two zones, which are fully saturated with the viscous fluid, and the flow in these zones is governed by the Brinkman equation. The space outside the sphere, where a clear fluid flows, is also divided into two zones: the Navier–Stokes zone and the Oseen flow zone. The solutions on the interface inside the sphere are matched with the condition proposed by Merrikh and Mohammad. The stream function in the Navier–Stokes zone is matched with that on the sphere surface by the condition proposed by Ochoa-Tapia and Whitaker. It is found that the drag on the spherical shell decreases as the permeability toward the sphere boundary increases.     

Flow past an axisymmetric body embedded in a saturated porous mediumÉcoulement derrière un corps à symétrie axiale plongé dans un milieu poreux saturé

The problem of viscous fluid past an axisymmetric body embedded in a fluid saturated porous medium is studied using the Brinkman's extension. A general formula for the drag on the body is derived in the form of a limit of an expression involving the stream function characterizing the flow. The flow past an axisymmetric approximate sphere is also considered. The stream function in this case is obtained in terms of Bessel functions and Gegenbauer's functions. The drag acting on the body is evaluated by using the formula derived. Its variation is studied with respect to geometric and permeability parameters. The special cases of flow past a sphere and a spheroid are obtained from the present analysis. To cite this article: D. Srinivasa Charya, J.V. Ramana Murthy, C. R. Mecanique 330 (2002) 417–423.     

Cell models for micropolar flow past a viscous fluid sphere

The Stokes axisymmetrical flow of an incompressible micropolar fluid past a viscous fluid sphere and the flow of a viscous fluid past a micropolar fluid sphere are investigated. The appropriate boundary conditions are taken on the surface of the sphere, while the proper conditions applied on the fictitious boundary of the fluid envelope vary depending on the kind of cell-model. These problems are solved separately in an analytical fashion, and the velocity profile and the pressure distribution inside and outside of the droplet are shown in several graphs for different values of the parameters. Numerical results for the normalized hydrodynamic drag force acting, in each case, on the spherical droplet-in-cell are obtained for various values of the parameters representing volume fraction, the classical relative viscosity, the micropolarity and spin parameters are presented both in tabular and graphical forms. Results of the drag force are compared with the previous particular cases.     

Plane problem of uniform flow of a two-layer fluid past a circular cylinder

An explicit solution is found for the problem of uniform horizontal flow of a two-layer fluid of infinite depth past a circular cylinder. The cylinder axis is perpendicular to the flow. The problem is solved within a linear formulation. The solution of the problem is expressed in the form of rapidly converging series with coefficients determined from a recurrence relation. The first seven terms of the series yield the values of the hydrodynamic loads with a relative accuracy of 10^–6. The results are in good agreement with the known values for similar problems in a homogeneous fluid. Tables of the lift and wave drag are given for homogeneous and two-layer fluids.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–97, January–February, 1996.     

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.     

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.     

A general formula for the drag on a sphere placed in a creeping unsteady micropolar fluid flow

In the present work, we investigate the creeping unsteady motion of an infinite micropolar fluid flow past a fixed sphere. The technique of Laplace transform is used. The drag formula is obtained in the physical domain analytically by using the complex inversion formula of the Laplace transform. The well known formula of Basset for the drag on a sphere placed in an unsteady viscous fluid flow and that of Ramkissoon and Majumdar for steady motion in the case of micropolar fluids are recovered as special cases. The obtained formula is employed to calculate the drag force for some micropolar fluid flows. Numerical results are obtained and represented graphically.     

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.     

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.     

Optimization of suction-blowing in a viscous fluid

The problem of drag minimization in a viscous fluid by means of controlled suction (blowing) is considered. In the low Reynolds number approximation matched asymptotic expansions are used to construct the second approximation and analytic solutions of the optimization problem are found for a sphere and a circular cylinder. Transition from unseparated to separated flow is accompanied by a qualitative restructuring of the optimal solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–32, May–June, 1989.     

Drag of a flat wedge in a two-phase flow

The drag of a flat wedge in a subsonic two-phase flow is investigated. In contrast to earlier work of Balanin and Zlobin [1] particular attention is devoted to the influence of the particle size. Detailed investigations are made of the dependences of the forces and aerodynamic coefficients on the concentration of the solid phase, the opening angle of the wedge, and the particle size. It is established that the drag coefficients depend on the particle size only for particles with diameters less than 30 um.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 177–180, March–April, 1982.     

Stokes flow past a porous spheroid embedded in another porous medium

A study is made with an analysis of an incompressible viscous fluid flow past a slightly deformed porous sphere embedded in another porous medium. The Brinkman equations for the flow inside and outside the deformed porous sphere in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in small parameter characterizing the deformation. The flow through the porous oblate spheroid embedded in another porous medium is considered as the particular example of the deformed porous sphere embedded in another porous medium. The drag experienced by porous oblate spheroid in another porous medium is also evaluated. The dependence of drag coefficient and dimensionless shearing stress on the permeability parameter, viscosity ratio and deformation parameter for the porous oblate spheroid is presented graphically and discussed. Previous well-known results are then also deduced from the present analysis.     

Transition flow of a fibrous suspension in a pipe as the flow of an anisotropic fluid

The transition flow is considered of a fibrous suspension in a pipe. The flow region consists of two subregions: at the center of the flow a plug formed by interwoven fibers and fluid moves as a rigid body; between the solid wall and the plug is a boundary layer in which the suspension is a mixture of the liquid phase and fibers separated from the plug [1–3]. In the boundary region the suspension is simulated as an anisotropic Ericksen—Leslie fluid [4, 5] which satisfies certain additional conditions. Equations are obtained for the velocity profile and drag coefficient of the pipe, which are both qualitatively and quantitatively in good agreement with the experimental results [6–8]. Within the framework of the model, a mechanism is found for reducing the drag in the flow of a fibrous suspension as compared to the drag of its liquid phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–98, September–October, 1985.     

Flow of a stream containing charged inertial particles past a conducting sphere

The problem of the flow of a hydrodynamic stream containing electrically charged particles past a conducting sphere is solved. The influence of the volume density of the electric charge and the potential of the sphere on the capture coefficient is determined for different values of the inertia parameter of the particles and different drag laws for them. It is shown that for an earthed sphere can appreciably exceed unity. In the formulation of the problem, it is assumed that the region of electrogasdynamic flow is bounded by two electrode grids, which simulate the exit of the source of charged particles and the surrounding electric conditions. The velocity field near the sphere is assumed irrotational. The problem is solved numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 21–26, July–August, 1982.I thank A. B. Vatazhin for assistance in the work.     

Circulation flow at the front surface of a sphere in a supersonic wake

Several theoretical and experimental studies of supersonic flow past a blunt body located in the wake behind another body have been made [1–7]. It has been shown that a reverse-circulation flow can occur in the shock layer at the front surface. The possibility of such a flow forming depends on the nonuniformity of the freestream flow and the Reynolds number. This paper presents new results of the theoretical study of the structure of the shock wave at the front surface of such a sphere, obtained on the basis of numerical solution of Navier-Stokes equations. It is shown that for a fixed nonuniformity of the freestream flow, an increase in the Reynolds number and cooling of the surface of the body lead to the formation of a secondary vortex in the region where the contour of the body intersects the axis of symmetry. A study is made of the variations of the drag and heat transfer parameters over the front surface of a cooled and thermally insulated sphere. The possibility of numerical simulation of the flow on the basis of the Euler equations is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–148, May–June, 1985.     

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.     

Flow past a sphere of two co-axial supersonic streams

We investigate the flow past a sphere of a parallel supersonic stream which is nonuniform in magnitude; such a flow can be considered as two co-axial streams of an ideal gas. The problem is solved numerically by the method of establishment [1]. Supersonic flow of nonuniform magnitude and direction past blunt bodies and a plane wall was investigated in [2–5],Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 89–94, September–October, 1970.The author wishes to thank G. F. Telenin for his attention to the paper.     

On turbulent separated flow past a truncated body of revolution

Results of testing a series of truncated bodies of revolution with convergent afterbodies in a hydrodynamic tunnel are presented. It is shown that the base pressure can be substantially raised and hence the total drag reduced by varying the shape and convergence of the afterbodies. This effect is caused by intense reverse jets formed as a result of the collision of flow particles moving toward the axis of symmetry.The turbulent flow past the bodies is calculated using the method of viscous-inviscid interaction. The formulas derived for the base pressure and drag coefficients agree satisfactorily with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–55, November–December, 1996.