Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

Thermal convection for a thermodependent herschel-bulkley fluid in an annular duct

This article presents a numerical and experimental investigation of the thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct under the conditions of a uniform heat flux density on the outer wall and an insulated inner wall. In the numerical analysis, it is assumed that: (i) the rheological behavior of the fluid can be expressed through the Herschel-Bulkley law: $\tau = \tau _s + K\dot \gamma ^n $ ; (ii) the flow is fully developed at the inlet; (iii) all fluid properties except consistency indexK are constant. TheK?T relation used isK=K ₀exp(?bT). The results obtained enable us to characterize completely the dynamical and thermal fields. The numerical solution is in good agreement with the experimental data, showing the reasonableness of the computed results.     

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     

Axial annular flow of plastic fluids: dead zones and plug-free flow

In axial annular flow, the shear stress decreases from its value τ(κR) at the inner cylinder to 0 at r = λR and increases from then on to τ(R) at the outer cylinder. For plastic fluids with a yield stress τ _c, λ will be such that flow commences when τ(κR) = τ(R) = τ _c. For fluids with position-dependent yield stresses (electro- and magnetorheological fluids are examples), the situation is more complex. While it is possible that yielding and flow occur everywhere, it is also possible that flow occurs only in parts of the fluid-filled space, and a dead zone (region in which the fluid is at rest) close to one of the walls exists. In that case, the fluid will flow no matter how small the applied pressure difference is. If P is large enough, the dead zone ceases to exist and flow without any plug is possible. The fluid flows as if no yield stress exists.

Experimental investigation of natural convection heat transfer in horizontal and inclined annular fluid layers

In the present study, an experimental investigation of heat transfer and fluid flow characteristics of buoyancy-driven flow in horizontal and inclined annuli bounded by concentric tubes has been carried out. The annulus inner surface is maintained at high temperature by applying heat flux to the inner tube while the annulus outer surface is maintained at low temperature by circulating cooling water at high mass flow rate around the outer tube. The experiments were carried out at a wide range of Rayleigh number (5 × 10⁴ < Ra < 5 × 10⁵) for different annulus gap widths (L/D _o = 0.23, 0.3, and 0.37) and different inclination of the annulus (α = 0°, 30° and 60°). The results showed that: (1) increasing the annulus gap width strongly increases the heat transfer rate, (2) the heat transfer rate slightly decreases with increasing the inclination of the annulus from the horizontal, and (3) increasing Ra increases the heat transfer rate for any L/D _o and at any inclination. Correlations of the heat transfer enhancement due to buoyancy driven flow in an annulus has been developed in terms of Ra, L/D _o and α. The prediction of the correlation has been compared with the present and previous data and fair agreement was found.     

Effect of acceleration on the wavy Taylor vortex flow

In this paper, we use a laser optical technique to investigate the characteristics of a wavy Taylor vortex flow between two concentric cylinders, with the inner cylinder subjected to a wide range of predetermined acceleration and the outer one at rest. We focus on the inner/outer radius ratio of 0.894, with an acceleration (dRe/dt*) from 0.1123 to 2,247, and Reynolds number from Re/Re _c =1.0 to 36. The results show that, with increasing Reynolds number, there is an initial increase in the wavelength of the wavy vortex flow (λ), and a decrease in the wave speed (c) before they asymptote to a constant value, which is a function of the acceleration. As for the wave amplitude (A), it is found that the effect of acceleration is significant only in a very narrow range of Reynolds numbers. Received: 21 August 2001 / Accepted: 22 November 2001     

Turbulent heat transfer in a flat plate boundary layer disturbed by a cylinder

Augmentation of heat transfer from a flat plate using a turbulence promoter has been studied. A circular cylinder 8 mm in diameter was placed in the turbulent boundary layer detached from the flat plate. It was located parallel to the plate and perpendicular to the flow direction. Clearance, c, between the cylinder and the flat plate was varied in nine steps: c=0, 1, 2, 3, 4, 6, 11, 20 and 29.5 mm. Measurements were made of the local heat transfer coefficients, mean velocity profiles, turbulence intensity profiles, static pressure and skin friction. Experimental results showed that the heat transfer deterioration which occurs just downstream of the cylinder at c=0 mm can be removed by displacing the cylinder a small distance from the wall. The improvement in heat transfer is mainly due to the unsteadiness of the recirculating flow on the plate and the effect of intense turbulence arriving at the near wall region from the lower shear layer of the cylinder wake. Heat transfer augmentation is most effective when c=4 mm and becomes less effective when c is increased more than 6 mm. The enhancement disappears far downstream from the 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.     

A rotating elliptic airfoil in fluid at rest and in a parallel freestream

This paper reports results of DPIV measurements on a two-dimensional elliptic airfoil rotating about its own axis of symmetry in a fluid at rest and in a parallel freestream. In the former case, we examined three rotating speeds (Re _c,Ω = 400, 1,000 and 2,000), and in the later case, four rotating speeds (Ro _c,Ω = 2.4, 1.2, 0.6 and 0.4), together with two freestream velocities (Re _c,u  = 200 and 1,000) and two starting configurations of the airfoil (i.e., chord parallel to (α ₀ = 0°) or normal (α ₀ = 90°) to the freestream). Results show that a rotating airfoil in a stationary fluid produces two distinct types of vortex structures depending on the Reynolds number. The first type occurs at the lowest Reynolds number (Re _c,Ω = 400), where vortices shed from the two edges or tips of the airfoil dissipated quickly, resulting in the airfoil rotating in a layer of diffused vorticity. The second type occurs at higher Reynolds numbers (i.e., Re _c,Ω = 1,000 and 2,000), where the corresponding vortices rotated together with the airfoil. Due to the vortex suction effect, the torque characteristics are likely to be heavily damped for the first type because of the rapidly subsiding vortex shedding, and more oscillatory for the second type due to persistent presence of tip vortices. In a parallel freestream, increasing the tip-speed ratio (V/U) of the airfoil (i.e., decreasing the Rossby number, Ro _c,Ω) transformed the flow topology from periodic vortex shedding at Ro _c,Ω = 2.4 to the generation of a “hovering vortex” at Ro _c,Ω = 0.6 and 0.4. The presence of the hovering vortex, which has not been reported in literature before, is likely to enhance the lift characteristics of the airfoil. Freestream Reynolds number is found to have minimal effect on the vortex formation and shedding process, although it enhances shear layer instability and produces more small-scale flow structures that affect the dynamics of the hovering vortex. Likewise, initial starting configuration of the airfoil, while affecting the flow transient during the initial phase of rotation, has insignificant effect on the overall flow topology. Unfortunately, technical constraint of our apparatus prevented us from carrying out complimentary force measurements; nevertheless, the results presented herein, which are more extensive than those computed by Lugt and Ohring (1977), will provide useful benchmark data, from which more advanced numerical calculations can be carried out to ascertain the corresponding force characteristics, particularly for those conditions with the presence of hovering vortex.     

T-RANS Simulations of Subcritical Flow with Heat Transfer Past a Circular Cylinder Surrounded by a Thin Porous Layer

We study flow and heat transfer to a cylinder in cross flow at Re = 3,900–80,000 by means of three-dimensional transient RANS (T-RANS) simulations, employing an RNG k − ε turbulence model. Both the case of a bare solid cylinder and that of a solid cylinder surrounded at some fixed distance by a thin porous layer have been studied. The latter configuration is a standard test geometry for measuring the insulating and protective performance of garments. In this geometry, the flow in the space between the solid cylinder and the porous layer is laminar but periodic, whereas the outer flow is transitional and characterized by vortex shedding in the wake of the cylinder. The results from the T-RANS simulations are validated against data from Direct Numerical Simulations and experiments. It is found that T-RANS is very well suited for simulating this type of flow. The transient nature of the flow underneath the porous layer is well reproduced, as well as the influence of vortex shedding on the heat transfer in the downstream stagnation zone. T-RANS results are found to be in much better agreement with DNS and experimental data than results from steady-state RANS.     

Heat transfer by laminar flow of an elastico-viscous liquid in a circular cylinder with linearly varying wall temperature

Summary The heat transfer by laminar flow of elastico-viscous liquids in a circular cylinder with linearly varying wall temperature has been studied by using the constitutive equation of motion for elastico-viscous liquids and energy equation. The flow phenomenon are characterized by two parameters R _c and S. The presence of the elastic elements in the viscous liquid considerably affects the velocity and temperature distributions.     

Flow around a cylinder surrounded by a permeable cylinder in shallow water

The change in flow characteristics downstream of a circular cylinder (inner cylinder) surrounded by an outer permeable cylinder was investigated in shallow water using particle image velocimetry technique. The diameter of the inner cylinder and the water height were kept constant during the experiments as d?=?50?mm and h _w ?=?25?mm, respectively. The depth-averaged free-stream velocity was also kept constant as U?=?170?mm/s which corresponded to a Reynolds number of Re_d?=?8,500 based on the inner cylinder diameter. In order to examine the effect of diameter and porosity of the outer cylinder on flow characteristics of the inner cylinder, five different outer cylinder diameters (D?=?60, 70, 80, 90 and 100?mm) and four different porosities (???=?0.4, 0.5, 0.6 and 0.7) were used. It was shown that both porosity and outer cylinder diameter had a substantial effect on the flow characteristics downstream of the circular cylinder. Turbulent statistics clearly demonstrated that in comparison with the bare cylinder (natural case), turbulent kinetic energy and Reynolds stresses decreased remarkably when an outer cylinder was placed around the inner cylinder. Thereby, the interaction of shear layers of the inner cylinder has been successfully prevented by the presence of outer cylinder. It was suggested by referring to the results that the outer cylinder having 1.6????D/d????2.0 and 0.4????D/d????0.6 should be preferred to have a better flow control in the near wake since the peak magnitude of turbulent kinetic energy was considerably low in comparison with the natural case and it was nearly constant for these mentioned porosities ??, and outer cylinder to inner cylinder diameter ratios D/d.     

Thermal characteristics in the annulus with an inner rotating rib-roughness cylinder

This work experimentally investigates the heat transfer characteristics in the annulus with an inner rotating rib-roughness cylinder, whose flow and thermal behaviors are associated with Taylor number (Ta) and centrifugal buoyancy parameter (Gr _Ω/Ta). The operating range of Ta is from 4.90 × 10² to 5.80 × 10⁵, while the surface of the inner cylinder is heated up with several constant heat fluxes (279, 425 and 597 W/m²) to obtain various values of Gr _Ω/Ta. Besides, three modes of the inner cylinder without/with longitudinal ribs are considered. The end of the annular channel is connected to a side chamber to fit practical applications (such as in the rotary blade coupling of a four-wheel-drive vehicle). The experimental results show that the average Nusselt number was almost constant at low Ta, but increased rapidly with Ta when Ta exceeded some critical value (3,000–5,200 for present study). Additionally, the Gr _Ω/Ta effect on the heat transfer was negligible herein. Furthermore, by comparing with the inner cylinder without longitudinal ribs, stalling ribs on the inner cylinder increases the transport of heat by a factor of 1.22 at 10⁵ < Ta < 10⁶, and embedding cavities into the ribs increases the transport of heat by a factor of 1.16 at 10⁵ < Ta < 10⁶. Finally, the relationships between the and the Ta for various modes of test sections were proposed.     

Computational flow and heat transfer of a row of circular jets impinging on a concave surface

A computational investigation is carried out to study the flow and heat transfer from a row of circular jets impinging on a concave surface. The computational domain simulates the impingement cooling zone of a gas turbine nozzle guide vane. The parameters, which are varied in the study include jet Reynolds number (Re _d = 5000–67800), inter-jet distance to jet diameter ratio (c/d = 3.33 and 4.67) and target plate distance to jet diameter ratio (H/d = 1, 3 and 4). The flow field, predicted with K-ω turbulence model and using Fluent 6.2.16, is characterized with the presence of a pair of counter rotating vortices, an upwash fountain flow and entrainment. The local pressure coefficient and Nusselt number variations along the concave plate are presented and these values are found to under predict the available experimental data by about 12%.     

Self-similar flow of a mixture of a non-ideal gas and small solid particles behind a shock wave in presence of heat conduction, radiation heat flux and a gravitational field

Similarity solutions are obtained for one-dimensional unsteady flow of a dusty gas behind a spherical shock wave with heat conduction and radiation heat flux under a gravitational field of heavy nucleus at the centre (Roche model). The dusty gas is assumed to be a mixture of small solid particles and a non-ideal gas. The equilibrium flow conditions are assumed to be maintained, and the heat conduction is expressed in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient α _R are assumed to vary with temperature and density. In order to obtain similarity solutions the density of the undisturbed medium is assumed to be constant. The effects of an increase in the value of the parameter of non-idealness of the gas in the mixture $\bar{b}$ , the mass concentration of the solid particles in the mixture K _p , the ratio of the density of the solid particles to the initial density of the gas G ₁ and the variation of the heat transfer parameters Γ _R and Γ _c are obtained.     

Finite element study of free surface turbulent flow in rotating channels

Under certain conditions of liquid flow through rotating channels, the Coriolis force can induce a free surface to be formed. This problem is of practical importance in a Coriolis wear tester, which is used for determining the sliding wear coefficient of wear materials in slurry handling equipment. A deforming Galerkin finite element method is presented for predicting two‐dimensional turbulent free surface mean flow in rotating channels. Reynolds‐averaged Navier–Stokes (RANS) equations are cast into weak(algebraic) form using primitive variables (velocity and pressure). Eddy viscosity is determined via a mixing length model. Velocity is interpolated biquadratically, while pressure is interpolated bilinearly. The kinematic condition is used to form the Galerkin residual for the free surface. The free surface is represented by Hermite polynomials of zeroeth order for continuity of position and slope. Combined Newton's iteration is used to simultaneously solve for the free surface and the field variables. Results of velocity and pressure fields, as well as the free surface are shown to converge with mesh‐size refinement. There is excellent respect for mass conservation. Results are presented for various values of Rossby number (Ro) and height‐based Reynolds number (Re_H). Parameter continuation in Ro and Re_H space is used to compute solutions at higher values of flow rate and angular velocity. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical modeling of multi micro jet impingement cooling of a three dimensional turbine vane

This paper reports a numerical investigation on the prediction of the thermal and hydrodynamic flow fields of multi micro jet impingement cooling of three dimensional turbine vanes. A three dimensional vane is modeled with an in-line array of impinging jets of diameters 0.5 and 0.25 mm. The numerical model consists of the steady, Reynolds-Averaged Navier–Stokes equations and the Kω SST Turbulence model. The governing equations are solved using a finite volume method. The crossflow mass velocity (G _c ) to jet mass velocity (G _j ) ratio, and the average and local heat transfer distributions are analyzed with varying mass velocity and jet-to-target spacing. It is found out that a significant decrease in crossflow ratio occurs with the smaller diameters. Due to the lower crossflow and higher exit velocities of the smaller jets, the penetration into the crossflow is much higher. Moreover, at a constant mass flow, the use of micro-jets enhanced the overall average heat transfer coefficient by 63%, while at a fixed pressure drop across the vane instead of the mass flow, the smaller diameters will still yield an enhancement of 34.3% in the overall average heat transfer coefficient.     

Stability of flow between two rotating cylinders in the presence of a constant heat flux at the outer cylinder and radial temperature gradient – wide gap problem

The stability of Couette flow of a viscous incompressible fluid between two concentric rotating cylinders in the presence of a radial temperature gradient due to a constant heat flux at the outer cylinder is studied. The critical values of `a' (the wave number) and Ta (the Taylor number) are listed in a table and some critical Taylor numbers are shown graphically. It is shown that as the heat flux is increased the flow becomes more unstable for all values of μ calculated, where μ is the ratio of the angular velocity of the outer cylinder to that of the inner cylinder. Received on 04 March 1997     

An experimental study of diabatic spiral vortex flow

In spiral vortex flow, between concentric cylinders with the inner cylinder rotating and the outer stationary, the addition of a thermal gradient across the gap is a known complicating factor. The present diabatic study for narrow and wide gaps (radius ratios N=0.955 and N=0.8), with a heated outer and adiabatic inner cylinder, was undertaken to investigate this problem. The heat transfer characteristics and the modes of transition have been investigated together with the relationship between them. Using standard on-line digital computer techniques, the onset of vortex flow and its higher transitions have been shown to cause a sharp increase in Nusselt number. At higher Taylor numbers, of the order of 10⁶, a marked change in the Nusselt number occurs with the onset of the transition to periodic turbulent vortex flow. Outer wall heating is seen to affect the modes of transition. Diabatic critical Taylor numbers are much higher than those for adiabatic conditions and are found to depend on the close approach of the vortices to the outer wall     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999