Free Convection in Confined Power-Law Fluids From Two Side-by-Side Cylinders in a Square Enclosure

Laminar free convection heat transfer in power-law fluids from two side-by-side cylinders (one hot and one cold) confined in a square duct has been studied numerically in the two-dimensional flow regime. For a fixed value of the ratio of cylinder radius to size of the enclosure, the effect of geometrical placement of the cylinders is studied on the resulting velocity and temperature fields in the laminar free convection regime by considering six asymmetric locations of the two cylinders. In particular, extensive results reported herein span the range of conditions of Grashof number, 10 to 10⁵; Prandtl number, 0.7 to 100, thereby yielding the range of the Rayleigh number as 7 to 10⁷; power-law index, 0.3 to 1.8; and the relative positions (dimensionless) of the cylinders with respect to the centerline, –0.25 to 0.25. The heat transfer characteristics are analyzed in terms of the local Nusselt number along the surfaces of the two cylinders and the enclosure walls. Overall, the average Nusselt number shows a positive dependence on both the Grashof number and the Prandtl number irrespective of the values of power-law index and relative positioning of the cylinders. Also, all else being equal, shear-thinning fluid behavior promotes heat transfer with reference to that in Newtonian fluids. When the two cylinders are situated close to the bottom wall, the rate of heat transfer is augmented with reference to that for the symmetric positioning of the cylinders along the horizontal mid-plane of the enclosure. Conversely, heat transfer deteriorates as the cylinders are located above the centerline of the enclosure. The present numerical results have been consolidated via the use of a modified Rayleigh number, thereby enabling the estimation of the average Nusselt number in a new application.     

Effect of Fluid Yield Stress on Natural Convection From Horizontal Cylinders in a Square Enclosure

In this study, laminar natural convection heat transfer to Bingham plastic fluids from two differentially heated isothermal cylinders confined in a square enclosure (with isothermal walls) has been investigated numerically. The governing partial differential equations have been solved over the ranges of the dimensionless parameters, namely, Rayleigh number, 10² to 10⁶, Prandtl number, 10 to 100, and Bingham number, 0.01 to 100, for seven locations of inner cylinders as ±0.25, ±0.2, ±0.1 and 0. These values correspond to the range of Grashof number varying from 10 to 10⁵. The detailed flow and temperature fields are visualized in terms of the streamlines and isotherm contours. Further insights are developed by examining the iso-shear rate contours and the yield surfaces delineating the fluid-like and solid-like regions. The corresponding heat transfer results are analyzed in terms of the distribution of the local Nusselt number along the cylinder surface together with its surface averaged value as functions of the Rayleigh number, Prandtl number, Bingham number, and positions of the cylinders. It is found that the average Nusselt number increases with the increasing values of the Rayleigh number and decreases with the increasing Bingham number. For sufficiently large values of the Bingham number, the average Nusselt number reaches its asymptotic value wherein heat transfer takes place solely by conduction. Based on the present numerical results, simple correlations for the prediction of the average Nusselt number and the limiting Bingham number have been developed. Also, a dimensionless criterion denoting the cessation of convection regime is outlined for this configuration.     

A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations

Numerical calculations are carried out for natural convection induced by a temperature difference between a cold outer square enclosure and a hot inner circular cylinder. A two-dimensional solution for unsteady natural convection is obtained, using the immersed boundary method (IBM) to model an inner circular cylinder based on the finite volume method for different Rayleigh numbers varying over the range of 10³–10⁶. The study goes further to investigate the effect of the inner cylinder location on the heat transfer and fluid flow. The location of the inner circular cylinder is changed vertically along the center-line of square enclosure. The number, size and formation of the cell strongly depend on the Rayleigh number and the position of the inner circular cylinder. The changes in heat transfer quantities have also been presented.     

Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations

In this model, a numerical study of two dimensional steady natural convection is performed for a uniform heat source applied on the inner circular cylinder in a square air (Pr = 0.7) filled enclosure in which all boundaries are assumed to be isothermal (at a constant low temperature). The developed mathematical model is governed by the coupled equations of continuity, momentum and energy and is solved by finite volume method. The effects of vertical cylinder locations and Rayleigh numbers on fluid flow and heat transfer performance are investigated. Rayleigh number is varied from 10³ to 10⁶ and the location of the inner cylinder is changed vertically along the centerline of the enclosure from − 0.25 L to 0.25 L upward and downward, respectively. It is found that at small Rayleigh numbers does not have much influence on the flow field while at high Rayleigh numbers have considerable effect on the flow pattern. In addition, the numerical solutions yield a two cellular flow field between the inner cylinder and the enclosure. Also, the total average Nusselt number behaves nonlinearly as a function of locations. Results are presented in terms of the streamlines, isotherms, local and average Nusselt numbers. Detailed results of the numerical has been compared with literature ones, and it gives a reliable agreement.     

Natural Convection in a Square Enclosure with Two Horizontal Cylinders

This study investigates natural convection in a cooled square enclosure with two inner heated circular cylinders with the same diameter. The centers of two equidiameter cylinders are placed at those of the lower and upper half of the enclosure, respectively. The immersed boundary method (IBM) to model the inner circular cylinders based on the finite volume method is used to study a two-dimensional natural convection for different Rayleigh numbers varying in the range of 10³ ≤ Ra ≤ 10⁵. The effect of the radius of inner circular cylinders in an enclosure on heat transfer and fluid flow at different Rayleigh numbers has been examined. As the Rayleigh number increases, the horizontal symmetry is broken and the asymmetry occurred from the smaller radius. As the radius decreases, the dependence of the convection on the Rayleigh number is considerable. The dependence of the Nusselt number on the radius and the Rayleigh number is presented.     

Laminar Natural Convection of Bingham Fluids in Square Cross-Sectioned Cylindrical Annular Cavity with Differentially Heated Vertical Walls Subjected to Constant Heat Fluxes

Steady-state numerical simulations have been conducted to investigate natural convection of yield stress fluids obeying Bingham model in square cross-sectioned axisymmetric cylindrical annular enclosure with vertical walls subjected to constant heat fluxes for nominal Rayleigh number range of 10³ to 10⁶, nominal Prandtl number of 10 to 10³ for different values of internal cylinder radius. It is found that the mean Nusselt number on the inner periphery increases (decreases) with increasing nominal Rayleigh (Bingham) number due to strengthening (weakening) of thermal advection. However, the values of the mean Nusselt number on the inner periphery obtained for Bingham fluids are smaller than that obtained for Newtonian fluids for the same set of nominal Rayleigh and Prandtl numbers. The mean Nusselt number normalized by the corresponding value obtained for pure conductive transport increases with increasing internal radius before asymptotically approaching the mean Nusselt number for a square enclosure. This suggests that the ratio of the convective to the conductive transport strengthens with increasing cylinder radius in the cylindrical annular cavity. Detailed physical explanations have been provided for the effects of the aforementioned parameters on the mean Nusselt number on the inner periphery and correlations have been proposed for the mean Nusselt number on the inner periphery for both Newtonian and Bingham fluids.     

Laminar Free Convection in Power-law Fluids in a Right Angle Triangular Duct with Heated Base

ABSTRACT

Laminar free convection in power-law fluids in a triangular duct is studied numerically to delineate the effects of the height-to-base ratio of the enclosure (0.2 to 2), power-law index (0.2 to 1.8), Grashof number (10 to 10⁴) and Prandtl number (0.7 to 100). The heat transfer is analyzed for the heated base with the other two walls being cold. Detailed kinematics is characterized by the formation of multiple recirculating zones ranging from two to four cells. Shear rate contours provide additional insights about the variation of the local viscosity in the fluid. Heatlines and the values of the Bejan number over the range of conditions are calculated to delineate the contributions of the entropy generation due to thermal effects and viscous dissipation. At low Grashof and/or Prandtl numbers, conduction dominates the overall heat transfer and this transition between the conduction and convection-dominated regimes is captured in terms of a modified Rayleigh number. The effect of aspect ratio on the Nusselt number is modulated by the values of Grashof and Prandtl numbers and power-law index. The present results have been consolidated via the use of a modified Rayleigh number for estimating the value of average Nusselt number in a new application.     

A Parametric Study of Prandtl Number Effects on Laminar Natural Convection Heat Transfer From a Horizontal Circular Cylinder to Its Coaxial Triangular Enclosure

A parametric study of Prandtl number effects on laminar natural convection heat transfer in a horizontal equilateral triangular cylinder with a coaxial circular cylinder is conducted. The Prandtl number is varied over a wide range from 10^?2 to 10⁵, which corresponds to a variety of working fluids. The governing equations with the Boussinesq approximation for buoyancy are iteratively solved using the finite volume approach. It is shown that the flow patterns and temperature distributions are unique for low-Prandtl-number fluids (Pr ≤ 0.1), and are nearly independent of Prandtl number when Pr ≥ 0.7. In addition, the inclination angle of the triangular enclosure is found to noticeably affect the variations of the local Nusselt number, and to have insignificant influence on the average Nusselt numbers for low Rayleigh numbers when Pr ≥ 0.7.     

Natural Convection in Power-Law Fluids in a Square Enclosure from Two Differentially Heated Horizontal Cylinders

Laminar natural convection has been numerically investigated from two differentially heated horizontal cylinders in a square enclosure filled with power-law fluids. Two basic configurations, namely, vertical- and diagonal-alignment of the cylinders at various locations have been considered. The coupled continuity, momentum and energy equations have been solved numerically to elucidate the effect of the Grashof number (10²–10⁴), Prandtl number (0.7–100) and power-law index (0.2–2) for a range of symmetric and asymmetric locations of the cylinders. The velocity and temperature fields are visualized in terms of streamlines, isothermal contours and plots of the local and average Nusselt number for different positions of the cylinders. The occurrence of the power-law index in the definitions of the Grashof and Prandtl numbers accentuates the interplay between the viscous, inertial and buoyancy forces thereby leading to nonlinearity in the observed trends. The presence of the dead zones coupled with the dominance of conduction under certain conditions strongly influences the overall heat transfer. All else being equal, it is possible to improve heat transfer for asymmetric positioning of the cylinders, especially at high values of the Prandtl number and Grashof number in shear-thinning fluids. A predictive correlation has been developed thereby enabling the estimation of the heat transfer coefficient in a new application in terms of the geometric and kinematic parameters.     

The structures of attractors reconstructed with time–evolution data of average heat transfer on thermal convection field in a vibrating square enclosure

In this paper the time‐dependent characteristics of surface‐averaged Nusselt number in a square enclosure with hot and cold side walls exposed to vertical vibrations were numerically examined. In the computation, the Prandtl number, the Rayleigh number, and the vibration Grashof number were held constant at 0.71, 10⁴, and 10⁶, respectively. The angular frequency of vibration was changed in the range between 10 and 7680. The results showed that the change in the characteristics of the surface‐averaged Nusselt number proposed by Fu and Shieh corresponded to the change in the shape of reconstructed attractor and that these regions could also be characterized by the three indices defined in phase space: average location of trajectory, the largest Lyapunov exponent, and the correlation dimension. Moreover, the time scale with which the autocorrelated coefficient of the surface‐averaged Nusselt number becomes 1/e was found to be a very important parameter for the time‐ and surface‐averaged Nusselt number. © 2000 Scripta Technica, Heat Trans Asian Res, 30(1): 11–21, 2001     

Magnetohydrodynamic Natural Convection in a Square Enclosure with Four Circular Cylinders Positioned at Different Rectangular Locations

We deploy a finite volume numerical computation to investigate the two-dimensional hydromagnetic natural convection in a cooled square enclosure in the presence of four inner heated circular cylinders with identical shape. The inner circular cylinders are placed in a rectangular array with equal distance away from each other within the enclosure and moving along the diagonals of the enclosure. All the walls of the enclosure are kept isothermal with temperatures less than that of the cylinders. A uniform magnetic field is applied along the horizontal direction normal to the vertical wall. All solid walls are assumed electrically insulated. Simulations are performed for a range of the controlling parameters such as the Rayleigh number 10³ to 10⁶, Hartmann number 0 to 50, and the dimensionless horizontal and vertical distance from the center of a cylinder to center of another cylinder 0.3 to 0.7. The study specifically aims to understand the effects of the location of the cylinders in the enclosure on the magnetoconvective transport, when they moved along the diagonals of the enclosure. It is observed that the unsteady behavior of the flow and thermal fields at relatively larger Rayleigh numbers and for some cylinder position are suppressed by imposition of the magnetic field. The heat transfer strongly depends on the position of the cylinders and the strength of the magnetic field. Hence, by controlling the position of the objects and the magnetic field strength, a significant control on the hydrodynamic and thermal transport can be achieved.     

Transient natural convective heat transfer of a low-Prandtl-number fluid from a heated horizontal circular cylinder to its coaxial triangular enclosure

Transient natural convective heat transfer of liquid gallium, which has a Prandtl number of 0.023 at 310 K, from a heated horizontal circular cylinder to its coaxial triangular enclosure is studied numerically by employing the control volume method. Two orientations of the triangular cylinder are investigated and the Grashof number is varied from 10⁴ to 10⁷. Development of natural convection is presented by means of the evolutions of the average Nusselt number over the outer triangular wall. Temporal stages during the course of development are identified and demonstrated through representative snapshots of streamlines and isotherms. The time-averaged Nusselt number is scaled with Grashof number for both conduction- and convection-dominated regimes. It is found that by placing horizontally the top side of the triangular cylinder, the convective flow becomes more stable and the overall heat transfer is enhanced. In addition, pitchfork bifurcation is explored quantitatively and its onset times are predicted as well.     

Buoyancy-induced flow and heat transfer in a porous annulus between concentric horizontal circular and square cylinders

ABSTRACT

This paper reports on natural convection heat transfer in a porous annulus between concentric horizontal circular and square cylinders. The heated inner circular cylinder is maintained at the uniform hot temperature T_h, whereas the cooled outer square duct is held at the uniform cold temperature T_c. A pressure-based collocated finite-volume method is used to numerically investigate the effects on the total heat transfer of Rayleigh number (Ra), Prandtl number (Pr), Darcy number (Da), porosity (?), and annulus aspect ratio (R/L). Results demonstrate that at low Ra values, conduction is the dominant heat transfer mode. Convection contribution to total heat transfer becomes more important beyond a critical Ra value, which decreases with an increase in Da and/or ?. Furthermore, an increase in the enclosure aspect ratio (R/L) leads to an increase in total heat transfer. A similar behavior is obtained with Prandtl number, where predictions indicate higher heat transfer rates at higher Pr values with its effect increasing as Ra increases. Streamlines and isotherms reveal flow separation for some of the reported cases. Limited computations are also performed for natural convection in a porous annulus between two horizontal concentric circular cylinders having the same inner and outer perimeters as the investigated enclosure. Comparison of the predicted average Nusselt number estimates with similar ones obtained in the original enclosure reveals a large percentage difference in values, demonstrating the strong influence of geometry on natural convection in enclosures.     

Three-dimensional natural convection in an enclosure with a sphere at different vertical locations

Numerical calculations are carried out for the three-dimensional natural convection induced by a temperature difference between a cold outer cubic enclosure and a hot inner sphere. The immersed-boundary method (IBM) to model a sphere based on the finite volume method is used to study a three-dimensional natural convection for different Rayleigh numbers varying in the range of 10³–10⁶. This study investigates the effect of the inner sphere location on the heat transfer and fluid flow. The flow and thermal fields eventually reach the steady state for all Rayleigh numbers regardless of the sphere location. For Rayleigh numbers of 10⁵ and 10⁶, the variation of local Nusselt number of the sphere along the circumferential direction is large, showing the strong three dimensionality of the natural convection in the enclosure unlike to the cases of lower Rayleigh numbers of 10³ and 10⁴. For the highest Rayleigh number, the local peaks of the Nusselt number on the top wall of the enclosure shows the sinusoidal distribution along the circumferential direction. The flow and thermal fields, and the local and surface-averaged Nusselt numbers on the sphere and the enclosure are highlighted in detail.     

Numerical study of mixed convection nanofluid in an annulus enclosure between outer rotating cylinder and inner corrugation cylinder

Numerical investigations are presented for mixed convection problems in a concentric inner sinusoidal cylinder and an outer rotating circular cylinder, which were kept at constant hot and cold temperatures, respectively. The free space between the cylinders and the enclosure walls was filled with a water‐Cu nanofluid. The governing equations are formulated for velocity, pressure, and temperature formulation and are modeled in COMSOL5.2a, a partial differential equation solver based on the Galerkin finite element method. The governing parameters considered are the solid volume fraction, [0, 0.02, 0.04, and 0.06], Re (1, 25, 100, 200, and 300), and Ra (less than 10⁴), and the inner cylinder corrugation frequencies varied from (N = 3, 6, and 9). According to the calculations, the Reynolds number, the Rayleigh number, the nanoparticle volume fraction, and the number of corrugations play an important role of forming the stream and isothermal lines, the local and the average Nusselt number inside the annulus enclosure. The average Nusselt number decreases with increasing Reynolds number and the number of corrugations, while it increases as the Rayleigh number and the volume fraction increase.     

NATURAL CONVECTION OVER A ROTATING CYLINDRICAL HEAT SOURCE IN A RECTANGULAR ENCLOSURE

A numerical study of laminar two-dimensional natural convection heat transfer from a uniformly heated horizontal cylinder rotating about its center, and placed in an isothermal rectangular enclosure, is performed using a spectral element method. The physical aspects of the flow and its thermal behavior are studied for a wide range of pure natural convection to mixed convection at low and high rotational speeds of the cylinder. The computer program has been validated against experimental correlations available on pure natural convection of heated bodies in enclosures. The rotation of the cylinder has been found to enhance the heat transfer. At low ratios of Rayleigh number to the square of the rotational Reynolds number, Ra / Re_ω ², the maximum temperature on the cylinder surface is decreased by as much as 25–35% from similar cases with fixed cylinders. At moderate values of Ra/ Re_ω ², the thermal plume rising above the cylinder is shifted in the rotation direction and the angular shift decreases as Ra / Re_ω increases. The rotation produces more uniform temperature and shear stress distributions around the cylinder surface. At high Rayleigh numbers the increase in rotation reduces the cylinder mean Nusselt number by 2–10% as compared with the fixed cylinder.     

Natural convection flow of Cu–Water nanofluid in horizontal cylindrical annuli with inner triangular cylinder using lattice Boltzmann method

In this paper the lattice Boltzmann method is used to investigate the effect of nanoparticles on natural convection heat transfer in two-dimensional horizontal annulus. The study consists of an annular-shape enclosure, which is created between a heated triangular inner cylinder and a circular outer cylinder. The inner and outer surface temperatures were set as hot (T_h) and cold temperatures (T_c), respectively and assumed to be isotherms. The effect of nanoparticle volume fraction to the enhancement of heat transfer was examined at different Rayleigh numbers. Furthermore, the effect of vertical, horizontal, and diagonal eccentricities at various locations is examined at Ra = 10⁴. The result is presented in the form of streamlines, isotherms, and local and average Nusselt number. Results show that the Nusselt number and the maximum stream functions increase by augmentation of solid volume fraction. Average Nusselt number increases when the inner cylinder moves downward, but it decreases, when the location of inner cylinder changes horizontally.     

Influences of Boundary Condition on Laminar Natural Convection of Bingham Fluids in Square Cross-Sectioned Cylindrical Annular Enclosures with Differentially Heated Vertical Walls

Numerical simulations have been carried out to analyze steady-state laminar natural convection of yield stress fluids obeying Bingham model in square cross-sectioned cylindrical annular enclosures with differentially heated vertical walls for both constant wall temperature and constant wall heat flux boundary conditions for active walls. The simulations have been performed under the assumption of axisymmetry for a nominal Rayleigh number range of 10³ to 10⁶ and nominal Prandtl number range of 10 to 10³ for different ratio of internal cylinder radius to cylinder height range of 0.125 to 16. The mean Nusselt number on the inner periphery for the constant wall heat flux configuration has been found to be smaller than that in the case of constant wall temperature configuration for a given set of values of nominal Rayleigh and Prandtl numbers for both Newtonian and Bingham fluid cases. The mean Nusselt number normalized by the corresponding value obtained for pure conductive transport increases with increasing internal radius before approaching the corresponding mean Nusselt number for square enclosures regardless of the boundary conditions. Detailed physical explanations have been provided for the effects of the aforementioned parameters on the mean Nusselt number on the inner periphery. Finally, the new Nusselt number correlations have been proposed for laminar natural convection of both Newtonian and Bingham fluids in square cross-sectioned cylindrical annular enclosures for both constant wall temperature and constant wall heat flux boundary conditions.     

Correlations for Natural Convective Heat Transfer From Vertical and Inclined Cylinders

Natural convective heat transfer from the exposed top surface of an inclined isothermal cylinder, with a circular cross section, mounted on a flat adiabatic base plate, has been numerically investigated. The cylinder is mounted normal to the flat adiabatic base plate. The numerical solution has been obtained by solving the dimensionless governing equations, subject to boundary conditions, using the commercial finite-volume method-based code FLUENT. The flow has been assumed to be symmetrical about the vertical center-plane through the cylinder. Results have only been obtained for Prandtl number of 0.7, which is the value existing in the application that originally motivated this study. The simulations consider Rayleigh numbers between 10³ and 10⁷, inclination angles between 0º and 180º, and dimensionless cylinder diameters between 0.25 and 1. The effects of dimensionless diameter, Rayleigh numbers, and inclination angles on the mean Nusselt number for the top exposed surface of the cylinder have been studied. Empirical correlations for the heat transfer rates from the top exposed surface of the cylinder have been derived.     

Numerical study of mixed convective heat transfer in a lid‐driven enclosure with rotating cylinders

A numerical simulation is performed to characterize the mixed convective transport in a three‐dimensional square lid‐driven enclosure with two rotating cylinders. The top wall is moving in the positive x‐direction, and the bottom wall is at a higher fixed temperature compared with all other isothermal walls. Both cylinders are rotating in its own plane about their centroidal axis. On the basis of rotation of both cylinders in clockwise or counter‐clockwise directions, four rotational models are studied. Various controlling parameters considered in the present study are Grashof number (10 ³ < Gr < 10 ⁵), rotating speed of the cylinder (5 < ω < 50), and the Reynolds number based on top wall movement is fixed to 100. The effect of cylinder rotation on the heat transfer of bottom wall is reported with the help of streamlines, contour plots of z‐component of vorticity, averaged and local Nusselt number, ratios of secondary flow and drag coefficient. It is observed that the heat transfer at the bottom wall is substantially dependent on the rotational model and rotational speed of the cylinder.