Effects of radial temperature gradient on the stability of viscous row in an annulus with a rotating inner cylinder

The Taylor stability problem is solved for the case of narrow gap between a rotating inner cylinder and a stationary outer cylinder where the latter is at a higher temperature than the former. The predictions are based on a numerical procedure with less restrictive assumptions than those previously adopted and they confirm the destablizing effect of a radial temperature gradient.

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Wirkung eines radialen Temperaturgradienten auf die Stabilität einer zähen Strömung in einem Kreisring, dessen innerer Zylinder rotiert

Zusammenfassung Das Problem der Taylor-Stabilität ist für den Fall eines engen Spaltes zwischen einem rotierenden inneren und einem festen äußeren Zylinder gelöst, wenn der äußere auf höherer Temperatur als der innere ist. Es wurde ein numerisches Verfahren gewählt, bei dem weniger einschränkende Voraussetzungen nötig waren als bisher. Die destabilisierende Wirkung eines radialen Temperaturgradienten wird bestätigt.

     

On the development of Taylor vortices in a vertical annulus with a heated rotating inner cylinder

A numerical study has been conducted to determine the heat transfer characteristics and flow patterns which develop around a rotating, heated vertical cylinder enclosed within a stationary concentric cylinder. A tall annulus (aspect ratio of 10) with fixed, adiabatic horizontal end-plates and a radius ratio of 0·5 has been considered. Furthermore, the effect that the introduction of buoyancy forces by heating the inner cylinder has on the development of the Taylor vortex flow is examined. It is observed that the formation of the Taylor vortices is delayed until the rotational parameter σ = Gr/Re² has a value below unity for any given Reynolds number Re which is above the critical value Re_crit for the formation of Taylor vortices in an isothermal flow. Also, the Taylor cells first appear at the top of the annulus. As σ is gradually decreased below unity, bifurcations to other states are observed. The final structure of the secondary flow is noticeably distorted in the mixed-convection mode, with the size of the Taylor cells varying greatly along the height of the annulus. This distortion diminishes as σ is further decreased, until the isothermal flow pattern is nearly recovered below σ = 0·01.     

Calculation of the turbulent characteristics of channel flow with rotating inner cylinder

An attempt to analyze the turbulent characteristics of flow in an annular channel with rotating inner cylinder is described which is based on the use of the pulsating-energy-balance equations in various directions of motion. The analytical results are compared with the experimental mean velocity distribution, the pulsation intensity, the correlation, and other data.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 121–127, May–June, 1969.     

Efficient computation of natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder

In this work, the natural convection in a concentric annulus between a cold outer square cylinder and a heated inner circular cylinder is simulated using the differential quadrature (DQ) method. The vorticity‐stream function formulation is used as the governing equation, and the coordinate transformation technique is introduced in the DQ computation. It is shown in this paper that the outer square boundary can be approximated by a super elliptic function. As a result, the coordinate transformation from the physical domain to the computational domain is set up by an analytical expression, and all the geometrical parameters can be computed exactly. Numerical results for Rayleigh numbers range from 104 to 106 and aspect ratios between 1.67 and 5.0 are presented, which are in a good agreement with available data in the literature. It is found that both the aspect ratio and the Rayleigh number are critical to the patterns of flow and thermal fields. The present study suggests that a critical aspect ratio may exist at high Rayleigh number to distinguish the flow and thermal patterns. Copyright © 2002 John Wiley & Sons, Ltd.     

Effects of radial temperature gradient on the stability of flow in a narrow-gap annulus with constant heat flux at the inner rotating cylinder

The stability of Couette flow of a viscous, imcompressible fluid between two concentric rotating cylinders in the presence of a radial temperature gradient due to constant heat flux at the inner cylinder is studied. Assuming the flow that takes place in a narrow gap, the eigenvalue problem is solved numerically. The critical values ofa (the wave number) andTa (the Taylor number) are listed in a table and these are also shown graphically in the case of the rotation of the cylinders in clockwise and counter-clockwise direction. It is observed that greater amount of heat flux has a stabilising effect.

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Einflüsse des radialen Temperaturgradienten auf die Stabilität einer Strömung in einem engen Ringspalt mit konstantem Wärmestrom am inneren rotierenden Zylinder

Zusammenfassung Es wird die Stabilität der Couette Strömung eines viskosen inkompressiblen Fluids zwischen zwei konzentrischen rotierenden Zylindern in Anwesenheit eines durch Wärmeabgabe vom inneren Zylinder hervorgerufenen Temperaturgradienten studiert. Unter der Annahme, daß die Strömung in einem engen Spalt vor sich geht, wird das Eigenwertproblem gelöst. Es werden die kritischen Werte vona (Wellenzahl) und vonTa (Taylorzahl) in Tabellen aufgelistet und diese Ergebnisse werden auch graphisch dargestellt für den Fall, daß die Zylinder gleichsinning und gegensinnig rotieren. Es wurde beobachtet, daß zunehmender Wärmestrom einen stabilisierenden Effekt hat.

     

On the orbital motion of a rotating inner cylinder in annular flow

In this paper, numerical calculations have been performed to analyse the influence of the orbital motion of an inner cylinder on annular flow and the forces exerted by the fluid on the inner cylinder when it is rotating eccentrically. The flow considered is fully developed laminar flow driven by axial pressure gradient. It is shown that the drag of the annular flow decreases initially and then increases with the enhancement of orbital motion, when it has the same direction as the inner cylinder rotation. If the eccentricity and rotation speed of the inner cylinder keep unchanged (with respect to the absolute frame of reference), and the orbital motion is strong enough that the azimuthal component (with respect to the orbit of the orbital motion) of the flow‐induced force on the inner cylinder goes to zero, the flow drag nearly reaches its minimum value. When only an external torque is imposed to drive the eccentric rotation of the inner cylinder, orbital motion may occur and, in general, has the same direction as the inner cylinder rotation. Under this condition, whether the inner cylinder can have a steady motion state with force equilibrium, and even what type of motion state it can have, is related to the linear density of the inner cylinder. Copyright © 2006 John Wiley & Sons, Ltd.     

Inertial waves in a rotating annulus with inclined inner cylinder: comparing the spectrum of wave attractor frequency bands and the eigenspectrum in the limit of zero inclination

We investigate theoretically inertial waves inside a liquid confined between two co-rotating coaxial cylinders of finite length. We consider the case of small viscosity and high angular velocity (i.e., small Ekman numbers), a parameter range of interest for many geophysical applications. In this case, inertial waves propagating in the container show multiple reflections at the walls before the waves can be damped by weak diffusion. We allow for the inner cylinder wall to be parallel or inclined with respect to the annulus’ vector of rotation (truncated cone). For the limit of zero viscosity, the wave propagation is governed by a boundary value problem that is composed of a linear second-order hyperbolic partial differential equation and the impermeability boundary conditions. For the special case of vertical cylinder walls (no inclination of the inner cylinder), this boundary value problem is separable, the corresponding eigenmodes can analytically be found and they are regular. However, when the inner cylinder wall is inclined, the hyperbolicity of the governing equation leads to internal shear layers (corresponding to singularities for the inviscid case). The geometrical structure of the shear layers can be explained by inertial waves, trapped on limit cycles denoted as wave attractors. The shape of the limit cycles depends on the wave frequency. In fact, the spectrum of regular modes, existing for the case of vertical cylinder walls, vanishes almost completely when the inner wall is inclined. Instead of a spectrum of discrete frequencies and regular eigenmodes, a spectrum of wave attractor frequency bands and singular eigenmodes exist. The question addressed here is whether the spectrum of wave attractor intervals collapses to the discrete frequency spectrum when the inclination angle of the inner cylinder goes to zero. To answer this question, the attractor frequency intervals are evaluated numerically for a series of decreasing cylinder inclination angles and are compared with the analytically found eigenspectrum for the case of zero inclination. Goal is to better understand the asymptotic behavior of the problem for decreasing inclination angles. This understanding helps to interpret results from laboratory experiments with geometries that differ from the perfect annulus with parallel cylinder walls.     

Factors affecting the stability of viscous axial flow in annuli with a rotating inner cylinder

Hot-wire measurements are presented of the onset of instability in developed axial flow and in both developing and developed tangential flow caused by inner cylinder rotation in concentric annuli of radius ratio N of 0.909, 0.809 and 0.565 for axial-flow Reynolds numbers (Re) between 86 and 2000. Within assessed uncertainty intervals, the consistency of marginal stability measurements, at four azimuthal locations 90° apart, indicates insensitivity to small variations in gap width; the measurements also confirm the destabilisation of nearly-developed and developed tangential flow identified by Takeuchi and Jankowski¹ with the occurrence at increasing Re of three-dimensional initial disturbances of spiral-vortex form. Comparison with earlier measurements suggests that in particular annuli, destabilisation may be delayed to higher Re by high values of certain geometrical factors, including radius ratio and the resultant end-effects parameter. Stability may also be restored or improved at high Re by reversion to developing tangential flow in which the initial instability is not of spiral-vortex form and where, for given N, the critical Taylor number appears uniquely related to the dimensionless axial co-ordinate. Stability is then generally greatest at low N.     

Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder

 In this paper the radial deformation and the corresponding stresses in a non-homogeneous hollow elastic cylinder rotating about its axis with a constant angular velocity is investigated. The material of the cylinder is assumed to the non-homogeneous and cylindrically orthotropic. The system of fundamental equations is solved by means of a finite difference method and the numerical calculations are carried out for the temperature, the components of displacement and the components of stress with the time t and through the thickness of the cylinder. The results indicate that the effect of inhomogeneity is very pronounced. Received on 21 December 2000     

Hydromagnetic convection in a rotating annulus with an azimuthal magnetic field

The problem of convection induced by radial buoyancy in an electrically conducting fluid contained by a rotating cylindrical annulus (angular frequency, ) in the presence of a homogeneous magnetic field (B) in the azimuthal direction is considered. The small gap approximation is used together with rigid cylindrical boundaries. The onset of convection occurs in the form of axial, axisymmetric or oblique rolls. The angle between the roll axis and the axis of rotation depends of the ratio between the Chandrasekhar number, QB², and the Coriolis number, . Fully three-dimensional numerical simulations as well as Galerkin representations for roll patterns including the subsequent stability analysis are used in the theoretical investigation. At finite amplitudes, secondary transitions to 3D-hexarolls and to spatio-temporal chaos are found. Overlapping regions of pattern stability exist such that the asymptotically realized state may depend on the initial conditions. PACS 47.27.-i, 47.65.+a     

Turbulence structure in an annuli with strongly heated inner cylinder

This structure of turbulent flow in an annulus with strong inner cylinder wall heating has been studied in terms of velocity and temperature with wall temperatures up to 707 °C and a Reynolds number of 48,000. With increase in wall heating, the turbulence very close to the wall was suppressed due to an increase in the kinematic viscosity. In the inner region, the intermittent mixing became intensive and the turbulent intensity increased whereas, in the outer region, the turbulence was suppressed since intermittent mixing was no longer effective. The results show that the thermal structure can be considered in terms of a passive scalar for wall temperatures less then around 200 °C, except in the leading region of the heated area. Received: 2 March 1998/Accepted: 2 November 1998     

Numerical and experimental investigation of thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct with rotating inner cylinder

Thermal convection for an incompressible Herschel-Bulkley fluid along an annular duct, whose inner cylinder is rotating and outer is at rest, is analyzed numerically and experimentally. The outer cylinder is heated at constant heat flux density and the inner one is assumed adiabatic. The first part of this study deals with the effect of the rheological behavior of the fluid and that of the rotation of the inner cylinder on the flow field and heat transfer coefficient. All the physical properties are assumed constant and the flow is assumed fully developed. The critical Rossby number Ro_c = (R₁Ω/U_d)_c, for which the dimension of the plug flow is reduced to zero is determined with respect to the flow behavior index, the radius ratio and the Herschel-Bulkley number for axial flow. The rotation of the inner cylinder induces a decrease of the axial velocity gradient at the outer cylinder thereby reducing the heat transfer between the heated wall and the fluid. The second part of this study introduces the variation of the consistency K with temperature and analyzes the evolution of the flow pattern and heat transfer coefficient along the heating zone. Two cases are distinguished depending on the Rossby number: (i) Ro < Ro_c, the plug flow dimension increases along the heating zone; (ii) Ro < Ro_c, the decrease of K with temperature leads to the reappearance of the plug flow. For high angular velocities, it is possible to have a plug zone attached to the outer cylinder. Finally, a correlation is proposed for the Nusselt number. It shows clearly that the effect of thermodependency of K on the heat transfer becomes more important with increasing rotational velocity of the inner cylinder.     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of inertia is introduced to account for the local inertia of the fluid.     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

The stability of the helical flow of pseudoplastic liquids in a narrow annular gap with a rotating inner cylinder

The stability of a laminar helical flow of pseudoplastic liquids in an annular gap with a rotating inner cylinder is investigated theoretically. The analysis is carried out under the assumption of a torroidal form of the secondary flow (torroidal Taylor vortices) for the narrow gap geometry. The power law model has been applied to describe the pseudoplasticity of liquids. The problem of the stability has been formulated with the aid of the method of small disturbances, and solved using the Galerkin method. In order to describe the stability limit the Reynolds and Taylor numbers defined with the aid of the mean viscosity value have been introduced. It has been found that pseudoplasticity has a considerably destabilizing influence on the Couette motion as well as on the helical flow in the initial range of the Reynolds number values (Re<30). A decrease of the flow index value,n, is accompanied by a decrease of the critical value of the Taylor number. This destabilizing effect of pseudoplasticity vanishes in the range of the larger values of the Reynolds number. In the rangeRe>30, the stability limit of the flow of pseudoplastic liquids can be described by a general dependence of the critical valueTa _c onRe, which is consistent with results obtained for the case of Newtonian fluids. a frequency number (Eq. (27)), 1/s - b wave number (Eq. (27)), 1/m - B = M/N parameter - d = R ₂ –R ₁ gap width, m - f(y, B, k) function of viscosity distribution (Eq. (7)) - f ₀ (x) function of viscosity distribution (narrow gap Eq. (35)) - F(x) = V(x)/V _m dimensionless distribution of axial flow velocity - G(x) = U(x) _i dimensionless distribution of angular flow velocity - K consistency coefficient, N sⁿ/m² - M = (P/L)R ₂ parameter of the stress field (Eq. (1)), N/m² - M ₀ torque per unit length, N - n flow index - N = M ₀/(2R ₂ ² ) parameter of the stress field (Eq. (1)), N/m² - p = 1/2n–1/2 parameter - pressure disturbance amplitude, N/m² - p pressure disturbance, N/m² - (P/L) pressure drop per unit length of the gap, N/m² - r radial coordinate, m - r _m location of the maximum value of the axial velocity, m - R ₁,R ₂ inner, outer radius of the annulus, m - Re = V _m 2d/ _m Reynolds number - S = (P/L · d/N) parameteer of the stress field (narrow gap) - t time, s - Ta = _i d ^3/2 R ₁ ^1/2 / _m Taylor number - U tangential velocity, m/s - U _i tangential velocity at the surface of the inner cylinder, m/s - V axial velocity, m/s - V _m mean axial velocity, m/s - V disturbance vector of velocity field, m/s - amplitude of theV _k -disturbance, m/s - X, Y, Z functions in Eqs. (36–38) - y = r/R ₂ dimensionless radial coordinate - x = (r—(R ₁+R ₂)/2)d radial coordinate (narrow gap) - L ₁ L ₄ linear operators in Eqs. (36–38) - = ad/V _m dimensionless frequency number - = b·d dimensionless wave number - component of the rate of strain tensor, 1/s - component of the rate of strain tensor corresponding to the disturbance, 1/s - = R ₁/R ₂ radius ratio - apparent viscosity, Ns/m² - ₀ apparent viscosity in the main flow, Ns/m² - µ disturbance of the apparent viscosity, Ns/m² - µ _m mean apparent viscosity, Ns/m² - density, kg/m³ - _ij component of the stress tensor, N/m² - angular velocity, rad/s - _i angular velocity of the inner cylinder, rad/s     

Transition from vortex to wall driven turbulence production in the Taylor–Couette system with a rotating inner cylinder

Axisymmetrically stable turbulent Taylor vortices between two concentric cylinders are studied with respect to the transition from vortex to wall driven turbulent production. The outer cylinder is stationary and the inner cylinder rotates. A low Reynolds number turbulence model using the k‐ω formulation, facilitates an analysis of the velocity gradients in the Taylor–Couette flow. For a fixed inner radius, three radius ratios 0.734, 0.941 and 0.985 are employed to identify the Reynolds number range at which this transition occurs. At relatively low Reynolds numbers, turbulent production is shown to be dominated by the outflowing boundary of the Taylor vortex. As the Reynolds number increases, shear driven turbulence (due to the rotating cylinder) becomes the dominating factor. For relatively small gaps turbulent flow is shown to occur at Taylor numbers lower than previously reported. Copyright © 2002 John Wiley & Sons, Ltd.     

Thermal analysis in swirl motion of Maxwell nanofluid over a rotating circular cylinder

In this paper, the mechanism of thermal energy transport in swirling flow of the Maxwell nanofluid induced by a stretchable rotating cylinder is studied. The rotation of the cylinder is kept constant in order to avoid the induced axially secondary flow. Further, the novel features of heat generation/absorption, thermal radiation, and Joule heating are studied to control the rate of heat transfer. The effects of Brownian and thermophoretic forces exerted by the Maxwell nanofluid to the transport ...     

Incompressible flow in a rapidly rotating cylinder with a source/sink distribution in the lateral wall and a free inner boundary

The flow of an incompressible fluid in a rapidly rotating right circular cylinder is considered. A source/sink mass distribution at the lateral wall, which is azimuthally uniform and symmetric across the midplane, causes a deviation from wheel flow. The container is only partially full and the inner free surface is allowed to deviate slightly from the vertical. A finite-difference solution of the full axisymmetric, non-linear governing equations was used to obtain the flow field. A special implicit technique for the Coriolis terms which maintains geostrophy was developed and is described. The results obtained for a low Rossby number flow compare quite favourably with the linearized solution. Results are also presented for a case wherein the non-linear terms are important.