Heat transfer by rotational flow in an annulus with porous lining

The flow and heat transfer in an annulus between rotating coaxial cylinders, with non-erodible porous lining, is investigated. The flow in the porous lining is obtained by using Brinkman equation. At the boundary between the porous lining and the free flow (the so called nominal surface), the velocity slip and the temperature slip are used. A quasi-numerical technique developed by the authors is employed in obtaining the solution of the energy equation. The effect of the thickness of the porous lining and the permeability on the velocity and the Nusselt numbers at the walls is studied.

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Wärmeübergang bei rotierender Strömung in einem Ring mit poröser Wand

Zusammenfassung In dieser Arbeit wird die Strömung und Wärmeübertragung zwischen rotierenden koaxialen Zylindern mit unauswaschbarem porösem Überzug untersucht. Die Strömung innerhalb des porösen Überzugs ist mit Hilfe der Brinkmanschen Gleichung berechnet. An der Grenze (der sogenannten Nominalfläche) zwischen dem Überzug und der freien Strömung wurde die Geschwindigkeitsgleitung und Temperaturgleitung benutzt. Die Energiegleichung ist mit Hilfe eines von den Autoren entwickelten quasi-numerischen Verfahrens gelöst. Der Einfluß der Dicke und der Durchlässigkeit des porösen Überzugs auf die Strömung und die Nusseltschen Zahlen an den Wänden wird untersucht.

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Nomenclature R ₂ radius of the outer cylinder forming the annulus - ₂ angular velocity of the outer cylinder - T ₂ temperature of the outer cylinder - R _l radius of the inner cylinder forming the annulus - ₁ angular velocity of the inner cylinder - T ₁ temperature of the inner cylinder - h thickness of the porous lining - R radial distance of any point in the annulus - V azimuthal component of velocity in zone 1 (of Fig. 1) - V part of velocity in zone 2 (of Fig. 1) due to transfer of momentum from the main flow - V _p velocity in zone 2 (of Fig. 1) - Q Darcy velocity in the porous medium (zone 2 of Fig. 1) - velocity slip parameter - k absolute permeability of the material used for lining - ₀ nondimensional shearing stress at the outer cylinder - _i nondimensional shearing stress at the inner cylinder - K thermal conductivity in zones 1 and 2 (of Fig. 1) - coefficient of viscosity of the fluid - T temperature in zone 1 (of Fig. 1) - T temperature in zone 2 (of Fig. 1) - temperature slip parameter - (Nu) _o nondimensional Nusselt number at the outer cylinder - (Nu) _i nondimensional Nusselt number at the inner cylinder - radii ratio - nondimensional rotational parameter - nondimensional thickness of the porous lining     

Direct numerical simulation of turbulent flows through concentric annulus with circumferential oscillation of inner wall

Periodic wall oscillations in the spanwise or circumferential direction can greatly reduce the friction drag in turbulent channel and pipe flows. In a concentric annulus, the constant rotation of the inner cylinder can intensify turbulence fluctuations and enhance skin friction due to centrifugal instabilities. In the present study, the effects of the periodic oscillation of the inner wall on turbulent flows through concentric annulus are investigated by the direct numerical simulation (DNS). The radius ratio of the inner to the outer cylinders is 0.1, and the Reynolds number is 2 225 based on the bulk mean velocity U_m and the half annulus gap H. The influence of oscillation period is considered. It is found that for short-period oscillations, the Stokes layer formed by the circumferential wall movement can effectively inhibit the near-wall coherent motions and lead to skin friction reduction, while for long-period oscillations, the centrifugal instability has enough time to develop and generate new vortices, resulting in the enhancement of turbulence intensity and skin friction.     

Instabilities of the Stewartson layer Part 2. Supercritical mode transitions

We investigate, both experimentally and numerically, the fluid flow in a spherical shell with radius ratio r_i/r_o=2/3. Both spheres rotate about a common axis, with _i>_o. The basic state consists of a Stewartson layer situated on the tangent cylinder, the cylinder parallel to the axis of rotation and touching the inner sphere. If the differential rotation is sufficiently large, non-axisymmetric instabilities arise, with the wavenumber of the most unstable mode increasing with increasing overall rotation. In the increasingly supercritical regime, a series of mode transitions occurs in which the wavenumber decreases again. The experimental and numerical results are in good agreement regarding this basic sequence of mode transitions, and the numerics are then used to study some of the finer details of the solutions that could not be observed in the experiment.     

Purely tangential flow of a PTT-viscoelastic fluid within a concentric annulus

An analytical solution is presented for the steady state, purely tangential flow of a viscoelastic fluid obeying the Phan-Thien–Tanner (PTT) constitutive equation in a concentric annulus with relative rotation of the inner and outer cylinders. The influence on the velocity distribution within the annulus and on fRe of the Weissenberg number, aspect ratio and an elongational parameter are investigated. The results show that the differences between the radial location of the minimum velocity and of the critical angular velocity compared with their Newtonian counterparts increase as the fluid elasticity increases. The results also show that fRe decreases with increasing Weissenberg number, radius ratio and the elongational parameter in the case of inner-cylinder rotation. In contrast, fRe increases with increasing radius ratio when the outer cylinder is rotating while the inner cylinder is at rest.     

The torsion of composite tubes and cylinders

Exact solutions are presented for the Saint-Venant torsion of circular tubes and solid cylinders which are reinforced by cylindrical inclusions of different material equally spaced around a concentric circle. The problems simulate those encountered in matrix rods reinforced by longitudinal fibers, and also in corresponding problems of reinforced concrete. Formulae are obtained for the boundary stress distributions and the torsional rigidities.Stress function formulations are made for the torsion of cylinders having multiply connected composite sections. Two systems of polar coordinates are employed, and use is made both of periodicity and symmetry. Three degenerate cases—the respective torsion of a homogeneous tube, ring of circular rods and tube with eccentric circular holes—are deduced for checking purposes. Several numerical examples are worked out and the results presented in tabular and graphical forms.     

The shape of a liquid surface between rotating concentric cylinders

Summary The shape of the free surface of a second order fluid, enclosed between vertical concentric cylinders with the inner cylinder rotating with angular velocity, is calculated. The calculation uses a perturbation procedure in which, for slow flows, the boundary conditions on the free surface are transformed to boundary conditions on the fixed horizontal surface obtained when is zero. The results are tested experimentally.With 3 figures     

Pressure-flow relationships for steady flow through an eccentric double circular tube

Pressure-flow relationships are derived for the steady flow through an eccentric double circular tube whose cross section takes the form of an annulus bounded by two circles which are in general not concentric. The velocity distribution can be obtained by solving a two-dimensional Poisson equation with the use of conformal mapping. In a case when the two circles touch, a mapping function of different type is required to solve the equation. The relationships thus derived are reduced to the well-known formula for the limit of concentric circles. It is found that when the region between touching circles becomes sufficiently thin, the tube conductance behaves like the cube of the difference of radii of the two circles.     

Stability of MHD Couette flow in a narrow gap annulus

A numerical solution to the MHD stability problem for dissipative Couette flow in a narrow gap is presented under following conditions: (i) the inner cylinder rotating with the outer one stationary, (ii) co-rotating cylinders, (iii) counter-rotating cylinders, (iv) an axially applied magnetic field, and (v) conducting and non-conducting walls.Results for the critical wave number and the critical Taylor number are presented and are compared with those of Chandrasekhar (1953). The agreement is very good. The amplitude of the radial velocity and the cell-pattern are shown on graphs for both the conducting and non-conducting walls and for different values of ± (=₂/₁, ₂-the angular velocity of the outer cylinder, ₁-the angular velocity of the inner cylinder) and Q the magnetic field parameter which is the square of the Hartman number. The effects of ± and Q on the stability of the flow are discussed. It is seen that the effect of the magnetic field is to inhibit the onset of instability, it being more so in the presence of conducting walls than in the presence of non-conducting walls.     

Unsteady circulatory flow about a circular cylinder with suction

Summary The complete Navier-Stokes equations which describe the unsteady flow of a viscous incompressible fluid when an infinite circular cylinder is given an impulsive twist, and simultaneously a constant suction velocity is imposed on the cylinder, are integrated using Laplace transforms. It is found that points which are at a greater distance from the cylinder are nearer to steadiness than points which are closer to the cylinder. Unsteady flow through a concentric annulus has also been considered.     

Velocity distributions in Taylor vortex flow with imposed laminar axial flow and isothermal surface heat transfer

Fully-developed flow in a concentric annulus formed by a stationary outer cylinder, which may be heated isothermally, and a rotatable inner cylinder has been studied experimentally by means of hot-wire anemometry techniques. Velocity profiles for the axial, tangential, and radial directions of flow have been obtained for adiabatic conditions and for wide and narrow annular gaps.It has been shown that the onset of Taylor vortex flow has a pronounced effect on the velocity profiles for all three directions. However, while the profiles for the axial and tangential directions are explicable, those for the radial direction are not so, at present. Also, it was found that heat transfer through the outer annular surface had a greater effect on the radial velocity profile than on the axial or tangential, but in the narrow gap case only.     

A method for correction of the wall-slip effect in a Couette rheometer

A simple method for correction of the wall-slip effect in a Couette rheometer was derived. The method requires only two series of measurements (two flow curves) performed in two measuring sets of any dimensions, and therefore it may be applied for the results obtained in each rheometer with a standard cup and bob set. The method was checked for experimental data and also verified theoretically for a hypothetical liquid. H height of cylinder - M torque - r distance from axis - R _i ,R ₀ radius of inner and outer cylinder - R _m average radius defined by Eq. (7) - u slip velocity - shear rate - shear rate for no-slip conditions - Newtonian viscosity - angular speed - angular speed of the rotating cylinder     

Flow structure in a flooded ball bearing

The flow structure in the confined space between the outer ring, the cage and the balls of a bearing is investigated using a large scale model allowing to perform visualizations, by tracer and dot-paint techniques, and velocity measurements, by Laser Doppler Velocity (LDV), through the transparent rotating outer ring. The visualization results show, in the region between two consecutive balls, the existence of a reversed flow on the cage surface resulting from the aspiration and blowing effect of the rotation of the balls in their cage housings. Systematic measurements of azimuthal velocities in different cross-sections of the gap confirmed the qualitative visualziation findings in laminar flow. For turbulent flow the results show that the extension of the reversed flow region is reduced and the reversed velocities are proportionally smaller as compared to the laminar case.List of symbols R radial position - R _b radius of the balls - R _c radius evaluated at the external surface of the cage - R _e radius evaluated at the inner wall of the outer cylinder - R _i radius evaluated at the outer wall of the inner cylinder - R _m radius of the center of the balls - Re ₀ Reynolds number in the space between the fixed inner cylinder and the rotating outer cylinder: Re ₀ = _e R _e(R _e - R _i)/v - Re ₁ Reynolds number in the space between the inner and outer cylinders: Re ₁ = 2_e R _e(R _e - R _i)/v - Re Reynolds number in the outer cylinder/cage gap: Re = _e R _e(R _e - R _c)/v - U axial velocity - V azimuthal velocity - V _e azimuthal velocity of the internal wall of the outer cylinder - V _i azimuthal velocity of the external wall of the inner cylinder - Z axial position - azimuthal position - kinematic viscosity - _i angular velocity of the inner cylinder - _e angular velocity of the outer cylinder - _c angular velocity of the balls about the axis of the bearing - _r angular velocity of the balls about their center This work was performed as part of a research effort aimed at investigating the many aspect of ball bearings flooded in cryogenic liquids and supported financially by the Centre National d'Etudes Spatiales (CNES) la Société Européenne de Propulsion (SEP) and the Centre National de la Recherche Scientifique (CNRS). The authors wish to deeply thank the many individuals, and in particular Dr. G. Jeanblanc from CNES and Mrs. Pierre and Moëllo from SEP, for their continuous encouragement.     

Circular Couette flow with pressure-driven axial flow and a porous inner cylinder

In a rotating filter separator a suspension is introduced at one end of the annulus between a rotating porous inner cylinder and a fixed impermeable outer cylinder. The filtrate is removed through the inner cylinder and the concentrate is removed from the opposite end of the annulus from which the suspension entered. The flow in a rotating filter separator is circular Couette flow with a pressure-driven axial flow and a suction boundary condition at the inner cylinder. Flow visualization was used to determine the effect of the Taylor number, axial Reynolds number, and radial Reynolds number on the types of flows present in the annulus. A rich variety of secondary vortical flows appear, depending upon the flow parameters. The radial inflow at the inner cylinder delays the appearance of supercritical circular Couette flow and prevents the appearance of certain flow regimes that have a helical vortex structure. Nevertheless, the average azimuthal velocity measured using laser Doppler velocimetry indicates that the velocity profile is nearly the same for all supercritical flow regimes.This work was supported by a grant from The Whitaker Foundation     

Flow of Robertson-Stiff fluids through an eccentric annulus

I.IntroductionInthepetroleumindustry,itisusuallythecasethatthedrillingstring(orcasingpipe)isnotlocatedinsidethecenteroftheflowgeometryduringdrillingandcompletillgpl'ocess,this,inturll,willaltertile'11owingbehaviorofdrillingmudandcementslurl.ieswhichtlowinginaneccentricallllulus.OilaccoUlltofviscousofnon-Newtoniantluid,flowillaneccentricannulusdifTeresmuchwiththatinacollcentricannulus.Mailyinvestigatorshaveconductedresearchworksonthisstlbject.Earlyin1935,TaoandDollovallrealizedthattheimpel.t…     

Ergodic Properties of Weak Asymptotic Pseudotrajectories for Semiflows

It is known that various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes can be analyzed by relating them to an appropriately chosen semiflow. Here, we introduce the notion of a stochastic process X being a weak asymptotic pseudotrajectory for a semiflow and are interested in the limiting behavior of the empirical measures of X. The main results are as follows: (1) the weak* limit points of the empirical measures for X axe almost surely -invariant measures; (2) given any semiflow , there exists a weak asymptotic pseudotrajectory X of such that the set of weak* limit points of its empirical measures almost surely equal the set of all ergodic measures for ; and (3) if X is an asymptotic pseudotrajectory for a semiflow , then conditions on that ensure convergence of the empirical measures are derived.     

Velocity measurements in a confined swirl driven recirculating flow

Laser-doppler measurements of the three components of mean velocity and their intensities have been made in an axisymmetric swirling isothermal turbulent combustion chamber flow. Swirl is generated by a gas-turbine aerodynamic swirler and the flow is representative of that found in the primary zone of many practical combustors. Two configurations were studied. In the first the fuel injector was removed and a central core jet entered the chamber via the resulting circular hole in the centre of the swirler. In the second case the injector was retained but a circular baffle was located at the exit plane: this latter device being necessary to prevent an inflow — not present for combusting flow — arising at the exit plane. The velocity data is sufficiently detailed to aid the testing and further development of methods for calculating combustion chamber flows.     

Effect of induced magnetic field on natural convection in vertical concentric annuli

In the present paper,we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field.The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account.The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity,induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli.The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables.It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder,while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder.These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases.It is also found that as the Hartmann number increases,there is a flattening tendency for both the velocity and the induced magnetic field.The influence of the induced magnetic field is to increase the velocity profiles.     

Wall visualization of swirling decaying flow using a dot-paint method

This paper deals with the visualization of swirling decaying flow in an annular cell fitted with a tangential inlet. A wall visualization method, the so-called dot-paint method, allows the determination of the flow direction on both cylinders of the cell. This study showed the complex structure of the flow field just downstream of the inlet, where a recirculation zone exists, the effects of which are more sensitive on the inner cylinder. The flow structure can be considered as three-dimensional in the whole entrance section. The swirl number and the entrance length were estimated using the measured angle of the streamlines. Experimental correlations of these two parameters, taking into account the Reynolds number and the axial distance from the tangential inlet, are given.List of symbols e = R ₂ – R ₁ thickness of the annular gap (m) - L _ax entrance length of axial flow on the outer cylinder (m) - L _ti length of the three-dimensional flow region on the inner cylinder (m) - L _to length of the three-dimensional flow region on the outer cylinder (m) - Q _v volumetric flowrate in the annular cell (m³s^–) - r radial position (m) - R ₁ external radius of the inner cylinder (m) - R ₂ internal radius of the outer cylinder (m) - Re=2eU _m /v Reynolds number - Sn swirl number - T time average resulting velocity (m s^–) - u time average axial velocity component (ms ^–) - average velocity in the annulus (m s^–) - w time average tangential velocity component (m s^–) - x axial location from the tangential inlet (m) - _e diameter of the tangential inlet (m) - streak angle with respect to the horizontal (degree) - angle with respect to the tangential inlet axis (degree) - gn kinematic viscosity of the working liquid (m²s^–)     

Feedback control of vortex shedding from a circular cylinder by cross-flow cylinder oscillations

Feedback control of vortex shedding from a circular cylinder in a uniform flow at moderate Reynolds numbers is studied experimentally with the cylinder subjected to feedback cylinder oscillations in cross-flow direction. The cylinder oscillation is digitally phase shifted with respect to the shedding vortex and is controlled by velocity feedback from the shear layer of the cylinder wake. Possible attenuation of vortex shedding is demonstrated by hot-wire measurements of the flow field and its mechanisms are studied by simultaneous data sampling and flow visualization with the smoke wire method and a laser-sheet illumination technique. Measurement results reveal substantial reduction in the fluctuating reference velocity at the optimum phase control. Flow visualization study indicates that the shear layer roll-up and the eventual vortex formation are dynamically attenuated under the control which results in a modification of the near wake.List of symbols A amplitude of cylinder oscillation - D cylinder diameter - E _u power spectrum function for fluctuating velocity u - frequency - R radius of circular cylinder - t time - u streamwise mean velocity - u streamwise fluctuating velocity - U streamwise mean velocity of main flow - u _r mean reference velocity - u _r fluctuating reference velocity - u _rf fluctuating reference velocity after filtering - y _c cylinder displacement - x, y, z coordinates from the cylinder center (Fig. 1) - feedback coefficient - phase lag The authors would like to express thanks to Professor K. Nagaya for his advice for designing electromagnetic actuators in the present experiments.