Effect of Prandtl Number on Free Convection from Two Cylinders in a Square Enclosure

This study explores the effect of Prandtl number on the laminar natural convection heat transfer to Newtonian fluids in a square enclosure consisting of one hot circular cylinder and one cold circular cylinder. The walls of the square enclosure are maintained isothermal and at the same temperature as the cold cylinder and the fluid medium. The governing partial differential equations have been solved numerically over the following ranges of conditions: Grashof number, 10 to 10⁵; Prandtl number, 0.7 to 100 (or the range of Rayleigh numbers as 7 to 10⁷); and relative positioning of the cylinders, ?0.25 to 0.25. However, the ratio of the radius of the cylinder to the side of the enclosure is held fixed at 0.2. Extensive results on the streamline and isotherm contours, the local Nusselt number distribution, and the average Nusselt number are discussed to delineate the influence of Grashof and Prandtl numbers on them for a given location with respect to the horizontal center line. The surface-averaged Nusselt number shows a positive dependence on Grashof and Prandtl numbers for a fixed location of the two cylinders. The heat transfer results have been correlated as a function of the Rayleigh number and geometric parameters, thereby enabling its prediction in a new application.     

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.     

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.     

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.     

Influences of a confined elliptic cylinder at different aspect ratios and inclinations on the laminar natural and mixed convection flows

Numerical investigations were carried out for natural and mixed convection within domains with stationary and rotating complex geometry by using an immersed-boundary method. The method was first validated with flows induced by natural convection in the annulus between concentric circular cylinder and square enclosure, and the grid-function convergence tests were also examined. Natural convection induced by isothermally elliptic cylinder was further investigated for different Rayleigh numbers within the range of 10⁴–10⁶ and the influence of the outer enclosure was also considered. The parameters investigated in the study included Rayleigh number, axis ratio and inclination angle of the elliptic cross-section. Local and average heat transfer characteristics were fully studied around the surfaces of both inner cylinder and outer enclosure. Finally, mixed convection in a square enclosure with an active rotating elliptic cylinder was considered and the heat transfer quantities of the system were obtained for different rotating speeds.     

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.     

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.     

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.     

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.     

Effects of Temperature-Dependent Viscosity on Natural Convection in Porous Media

This analysis has studied natural convection for the temperature-dependent viscosity of fluids inside porous media between two concentric spheres by numerically solving the Brinkman–Darcy–Forchheimer model, vorticity transport, and energy equations. Parameters included Rayleigh numbers (5.0 × 10³–8.0 × 10⁴) at radius ratios of 1.5, 2.0, and 3.5 with porosities of 0.4 and 0.9 for variable-viscosity fluids with Prandtl numbers (158, 405, and 720) when the Darcy number was changed at 0.1 and 0.001. The results showed that the mean Nusselt number varied with Rayleigh number, porosity, radius ratio, and variable viscosity but did not change with the Darcy number.     

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.     

Natural convection heat transfer of high temperature gas in an annulus between two vertical concentric cylinders

Water cooling panels have been adopted as the vessel cooling system of the High Temperature Engineering Test Reactor (HTTR) to cool the reactor core indirectly by natural convection and thermal radiation. In order to investigate the heat transfer characteristics of high temperature gas in a vertical annular space between the reactor pressure vessel and cooling panels of the HTTR, we carried out experiments and numerical analyses on natural convection heat transfer coupled with thermal radiation heat transfer in an annulus between two vertical concentric cylinders with the inner cylinder heated and the outer cylinder cooled. In the present experiments, Rayleigh number based on the height of the annulus ranged from 2.0 × 10⁷ to 5.4 × 10⁷ for helium gas and from 1.2 × 10⁹ to 3.5 × 10⁹ for nitrogen gas. The numerical results were in good agreement with the experimental ones regarding the surface temperatures of the heating and cooling walls. As a result of the experiments and the numerical analyses, the heat transfer coefficient of natural convection coupled with thermal radiation was obtained as functions of Rayleigh number, radius ratio, and the temperatures and emissivities of the heating and cooling wall surfaces. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 34(5): 293–308, 2005; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20070     

Effects of Wall-Located Heat Barrier on Conjugate Conduction/Natural-Convection Heat Transfer and Fluid Flow in Enclosures

The effects of a heat barrier, located in the ceiling wall of an enclosure, on conjugate conduction/natural convection are investigated numerically. The vertical walls of the enclosure are differentially heated and the horizontal walls are adiabatic. Heatline technique is used to visualize heat transport. The variations of average Nusselt number, dimensionless heat transfer rate through the ceiling wall, and dimensionless overall heat transfer rate are studied. Calculations are performed for different Rayleigh numbers (10³ ≤ Ra ≤ 10⁶), thermal conductivity ratios (1 ≤ K ≤ 100), dimensionless locations of the heat barrier (0 < X _h  < 1),and two dimensionless ceiling wall thicknesses (D = 0.05 and D = 0.20). For high thermal conductivity ratio (K = 100), the heat barrier considerably reduces the dimensionless overall heat transfer rate. The effect of the heat barrier on dimensionless heat transfer rate through the enclosure increases as the Rayleigh number decreases. For low Rayleigh number (i.e., Ra = 10³), a location exists in the ceiling wall for which the dimensionless overall heat transfer rate is minimum.     

NUMERICAL EVALUATION OF WEAKLY TURBULENT FLOW PATTERNS OF NATURAL CONVECTION IN A SQUARE ENCLOSURE WITH DIFFERENTIALLY HEATED SIDE WALLS

This article presents the results of numerical evaluation of weakly turbulent natural convection of air in a rectangular enclosure with differentially heated side walls and adiabatic horizontal walls. The turbulence in the natural convection was described by k–ε equations, which were solved by Strang splitting, while average thermal and fluid flow fields were described by statistically averaged equations, which were solved by the projection method PmIII. The combined application of projection method and the Strang splitting characterizes the numerical method in this study. Numerical results for Rayleigh number 1.58 × 10⁹ have revealed reasonable agreement with the existing experimental ones, with some discrepancy attributable to the adiabatic boundary conditions on the horizontal walls. The results are also in good agreement with some published numerical results, particularly at higher Rayleigh numbers. However, comparison with the latest experimental data reveals that the turbulent heat flux model is not quite capable of giving satisfactory temperature distribution.     

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.     

双绝热圆柱对水平三角腔内自然对流影响的数值研究

针对含双绝热圆柱的底部加热水平等腰三角腔内空气的稳态层流自然对流开展研究.通过有限容积法对控制方程进行了数值求解,其中瑞利数的变化范围为104 ～106,圆柱体的尺寸比则分别为0（无圆柱体）、1/8和1/4.基于计算结果对自然对流的流动与传热特性随瑞利数和尺寸比的变化规律进行了分析和讨论.结果表明,双绝热圆柱的存在较大程度上改变了三角腔内自然对流的流型和温度场分布,但对整体传热影响较小,仅略微提高了平均努赛尔数,强化传热的效果在尺寸比为1/8时较为明显.     

Natural Convection Heat Transfer in an Enclosure Filled with an Ethylene Glycol—Copper Nanofluid Under Magnetic Fields

This article presents the results of a numerical study on natural convection in a square enclosure filled with ethylene glycol–copper nanofluid in the presence of magnetic fields. Two opposite horizontal walls of the enclosure are insulated and the two vertical walls are kept constant at different temperatures. A uniform horizontal magnetic field is externally imposed. The governing equations (mass, momentum, and energy) are formulated and solved numerically with a finite element using COMSOL Multiphysics. The effects of pertinent parameters such as Rayleigh number (10³ ≤ Ra ≤ 10⁷), Hartmann number (0 ≤ Ha ≤ 120), and solid volume fraction (0 ≤ φ ≤ 0.06) on the flow and the heat transfer performance of the enclosure are examined when the Prandtl number is assumed to be Pr = 151.     

Natural convection between a circular enclosure and an elliptic cylinder using Control Volume based Finite Element Method

In this study natural convection heat transfer in a cold outer circular enclosure containing a hot inner elliptic cylinder is investigated numerically using the Control Volume based Finite Element Method (CVFEM). Both of the circular enclosure and the inner cylinder are maintained at constant temperatures with air filled inside the enclosure. The governing equations are used in their vorticity stream function form to simulate the fluid flow and heat transfer. The numerical calculations are performed for various Rayleigh numbers, the inclination angle of the enclosure and different sizes of inner cylinder. The results show that streamlines, isotherms, and the number, size and formation of the cells inside the enclosure are strongly depend on these parameters which considerably enhance the heat transfer rate.     

Numerical Simulation of Buoyancy-Induced Turbulent Flow Between Two Concentric Isothermal Spheres

Buoyancy-induced turbulent flow and natural convection heat transfer between two differentially heated concentric isothermal spheres is studied numerically. The low-Reynolds-number k–ω model is used for turbulence modeling. The two-dimensional governing equations are discretized using control volume method and solved by employing the alternating direction implicit scheme. Results are presented in the form of streamline and temperature patterns, and local and average Nusselt numbers, over the heated and cooled boundaries for a wide range of Rayleigh numbers (10²–10¹⁰), extending the previous studies to the turbulent flow regime and for the radius ratio of 2. The results of the flow pattern and average Nusselt numbers were compared with the previously published experimental and numerical investigations and very good agreements were observed. For low values of Rayleigh numbers, regions with conduction-dominated flow pattern accompanied with low values of Nusselt numbers were observed, while for higher Rayleigh numbers, the flow pattern was changed to the convection dominated boundary layer type flow, resulting in an increase in the rate of heat transfer and flow velocities adjacent to both inner and outer boundaries. The average Nusselt numbers were correlated against Rayleigh number and a 1/4 power dependence of Ra in both laminar and turbulent regimes is obtained.     

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.