Effects of thermal boundary conditions on natural convection flows within a square cavity

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁵ and Prandtl number Pr, 0.7 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Laminar free convection from a horizontal semi-circular cylinder to power-law fluids

Extensive numerical results on the flow and thermal fields are presented for free convection from a semi-circular cylinder (flat base upward) immersed in quiescent power-law fluids for the following ranges of conditions: Grashof number, 10 ? Gr ? 10⁵, Prandtl number, 0.72 ? Pr ? 100, and power-law index, 0.2 ? n ? 1.8. The heat transfer characteristics are analyzed in terms of the isotherm patterns, local and average Nusselt number as functions of the pertinent dimensionless parameters. The flow field is visualized in terms of the streamline patterns adjacent to the surface of the cylinder for a range of values of the Grashof number, Prandtl number and power-law index. A separated flow region forms at as low values of the Prandtl number as Pr = 0.72 for n ? 1 (Newtonian and shear-thickening fluids); whereas for shear-thinning fluids (n < 1), the flow remains attached to the cylinder surface over the range of conditions encompassed here. The bubble size grows with Grashof number and it shrinks with Prandtl number. In order to quantify the deviation from the Newtonian behaviour, the normalized values of average Nusselt number are analyzed as a function of the power-law index. In addition, a correlation is proposed for average Nusselt number as a function of the Grashof number, Prandtl number and power-law index. In general terms, shear-thinning fluid behaviour enhances heat transfer whereas shear-thickening has adverse influence on it.     

Prandtl number dependence of laminar natural convection heat transfer in a horizontal cylindrical enclosure with an inner coaxial triangular cylinder

A numerical study of laminar convection heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure is performed to investigate the Prandtl number effect on flow and heat transfer characteristics. The Prandtl number over several orders of magnitude (10^?2 < Pr < 10³) as well as different aspect ratios (AR = 1.2 and 2.0) and different Rayleigh numbers (Ra = 10³, 10⁴, 10⁵, and 10⁶) are considered. The finite volume approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The computed flow patterns and temperature fields are shown by means of streamlines and isotherms, respectively, and the local and average heat transfer coefficients are also presented. It is found that the flow and heat transfer characteristics for a low Prandtl number fluid (Pr = 0.03) are unique and they are almost independent of Prandtl number when Pr ? 0.7. The entire spectrum of Prandtl number investigated can be divided into three sections based on the variations of average heat transfer coefficients. In each section, correlating equations of the average Nusselt number to the Rayleigh number are proposed with the maximum deviation less than 3%.     

Numerical investigation of several physical and geometric parameters in the natural convection into trapezoidal cavities

Natural convection in trapezoidal cavities, especially those with two internal baffles in conjunction with an insulated floor, inclined top surface, and isothermal left-heated and isothermal right-cooled vertical walls, has been investigated numerically using the Element based Finite Volume Method (EbFVM). In numerical simulations, the effect of three inclination angles of the upper surface as well as the effect of the Rayleigh number (Ra), the Prandtl number (Pr), and the baffle’s height (H_b) on the stream functions, temperature profiles, and local and average Nusselt numbers has been investigated. A parametric study was performed for a wide range of Ra numbers (10³ ? Ra ? 10⁶) H_b heights (H_b = H¹/3, 2H¹/3, and H¹), Pr numbers (Pr = 0.7, 10 and 130), and top angle (θ) ranges from 10 to 20. A correlation for the average Nusselt number in terms of Pr and Ra numbers, and the inclination of the upper surface of the cavity is proposed for each baffle height investigated.     

Natural convection effects on heat and mass transfer in a curvilinear triangular cavity

Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (?10 ? Br ? 10), Lewis number (0.1 ? Le ? 20) and thermal Rayleigh number (10⁴ ? Ra_T ? 10⁶). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from ?10 to 20 at Ra_T = 10⁶. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number.     

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in power-law fluids in the steady flow regime

The continuity, momentum and energy equations describing the flow and heat transfer of power-law fluids over a semi-circular cylinder have been solved numerically in the two-dimensional steady flow regime. The influence of the Reynolds number (Re), Prandtl number (Pr) and power-law index (n) on the local and global flow and heat characteristics have been studied over wide ranges of conditions as follows: 0.01 ? Re ? 30, 1 ? Pr ? 100 and 0.2 ? n ? 1.8. The variation of drag coefficient and Nusselt number with the Reynolds number, Prandtl number and power-law index is shown over the aforementioned ranges of conditions. In addition, streamline and isotherm profiles along with the recirculation length and distribution of pressure coefficient and Nusselt number over the surface of the semi-circular cylinder are also presented to gain further insights into the nature of the underlying kinematics. The wake size (recirculation length) shows almost linear dependence on the Reynolds number (Re ? 1) for all values of power-law index studied herein. The drag values show the classical inverse variation with the Reynolds number, especially for shear-thinning fluids at low Reynolds numbers. The point of maximum pressure coefficient is found slightly displaced from the front stagnation point for highly shear-thinning fluids, whereas for shear-thickening and Newtonian fluids, it coincides with the front stagnation point. For fixed values of the Prandtl number and Reynolds number, the rate of heat transfer decreases with the gradual increase in power-law index; this effect is particularly striking at high Prandtl numbers due to the thinning of the thermal boundary layer. Conversely, as expected, shear-thinning behavior facilitates heat transfer and shear-thickening impedes it. The effect of power-law index on both momentum and heat-transfer characteristics is seen to be appreciable at low Reynolds numbers and it gradually diminishes with the increasing Reynolds number.     

Jet impingement cooling of a horizontal surface in an unconfined porous medium: Mixed convection regime

Two-dimensional slot jet impingement cooling of an isothermal horizontal surface immersed in an unconfined porous medium is simulated numerically to gain insight into thermal characteristics under mixed convection conditions with the limitation of the Darcy model. The jet direction is considered to be perpendicular from the top to the horizontal heated element; therefore, the jet flow and the buoyancy driven flow are in opposite directions. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 0.5), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. It is shown that mixed convection mode can cause minimum average Nusselt number at two values of Péclet number and a maximum average Nusselt number occurs in between theses two Péclet numbers at higher Rayleigh number due to counteraction of jet flow against buoyancy driven flow. Hence careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10⁻⁵ irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10⁻⁴. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Two-dimensional unsteady forced convection heat transfer in power-law fluids from a cylinder

Forced convection heat transfer characteristics of a cylinder (maintained at a constant temperature) immersed in a streaming power-law fluids have been studied numerically in the two-dimensional (2-D), unsteady flow regime. The governing equations, namely, continuity, momentum and thermal energy, have been solved using a finite volume method based solver (FLUENT 6.3) over wide ranges of conditions (power law index, 0.4 ? n ? 1.8; Reynolds number, 40 ? Re ? 140; Prandtl number, 1 ? Pr ? 100). In particular, extensive numerical results elucidating the influence of Reynolds number, Prandtl number and power-law index on the isotherm patterns, local and average Nusselt numbers and their evolution with time are discussed in detail. Over the ranges of conditions considered herein, the nature of flow is fully periodic in time. The heat transfer characteristics are seen to be influenced in an intricate manner by the value of the Reynolds number (Re), Prandtl number (Pr) and the power-law index (n). Depending upon the value of the power-law index (n), though the flow transits from being steady to unsteady somewhere in the range ～33 < Re < 50, the fully periodic behavior is seen only beyond the critical value of the Reynolds number (Re). As expected, the average Nusselt number increases with an increase in the values of Reynolds and/or Prandtl numbers, irrespective of the value of the flow behavior index. A strong influence of the power-law index on both local and time-averaged Nusselt numbers was observed. Broadly, all else being equal, shear-thinning behavior (n < 1) promotes heat transfer whereas shear-thickening behavior (n > 1) impedes it. Furthermore, this effect is much more pronounced in shear-thinning fluids than that in shear-thickening fluids.     

Steady natural convection flows in a square cavity with linearly heated side wall(s)

The present numerical study deals with natural convection flow in a closed square cavity when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Non-linear coupled PDEs governing the flow have been solved by penalty finite element method with bi-quadratic rectangular elements. Numerical results are obtained for various values of Rayleigh number (Ra) (10³ ⩽ Ra ⩽ 10⁵) and Prandtl number (Pr) (0.7 ⩽ Pr ⩽ 10). Results are presented in the form of streamlines, isotherm contours, local Nusselt number and the average Nusselt as a function of Rayleigh number.     

Unsteady wake dynamics and heat transfer in forced and mixed convection past a circular cylinder in cross flow for high Prandtl numbers

This paper investigates the combined effect of Prandtl number and Richardson number on the wake dynamics and heat transfer past a circular cylinder in crossflow using a SUPG based finite element method. The computations are carried out for 80 < Re < 180, 0.7 < Pr < 100 and $0?\mathrm{Ri}?2$. The results have been presented for both forced and mixed convection flows. In the case of forced convection, crowding of temperature contours with reduced spatial spread is observed for increasing Prandtl numbers. The local and average Nusselt numbers are found to increase with increasing Reynolds number and Prandtl number. The average Nusselt number and Colburn factor are found to vary as Re^0.548 Pr ^0.373 and Re^?0.452, respectively. The extrapolated results of the average Nusselt number for low and high Reynolds numbers are found to match quite well with the available results in literature. Effect of Prandtl number shows various interesting phenomena for the mixed convective flows. Increasing the Prandtl numbers resulted in decreasing deflection and strength in the wake structures. The effect of baroclinic vorticity production during vortex shedding has been demonstrated at the vicinity of the cylinder and near wake. The Strouhal number is found to decrease with increasing Prandtl number, in the case of buoyancy induced flow. The effect of increasing Prandtl number is manifested as the stabilizing effect in the flow. This is, perhaps, the first time that such behavior for the Prandtl number is being reported. Additionally it is observed that the average Nusselt number decreases with increasing Richardson number.     

Flow over and forced convection heat transfer in Newtonian fluids from a semi-circular cylinder

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in unconfined flowing Newtonian fluids have been investigated numerically. The governing equations, namely, continuity, Navier–Stokes and energy, have been solved in the steady flow regime over wide ranges of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr ? 100). Prior to the investigation of drag and heat transfer phenomena, the critical values of the Reynolds number for wake formation (0.55 < Re^c < 0.6) and for the onset of vortex shedding (39.5 < Re_c < 40) have been identified. The corresponding values of the lift coefficient, drag coefficient, and Strouhal number are also presented. After establishing the limit of the steady flow regime, the influence of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr = 0.72, 1, 10, 50 and 100) on the global flow and heat transfer characteristics have been elucidated. Detailed kinematics of the flow is investigated in terms of the streamline and vorticity profiles and the variation of pressure coefficient in the vicinity of the cylinder. The functional dependence of the individual and total drag coefficients on the Reynolds number is explored. The Nusselt number shows an additional dependence on the Prandtl number. In addition, the isotherm profiles, local Nusselt number (Nu_L) and average Nusselt number (Nu) are also presented to analyze the heat transfer characteristic of a semi-circular cylinder in Newtonian media.     

Boundary layer instability of the natural convection flow on a uniformly heated vertical plate

Direct numerical simulation is employed to investigate the two-dimensional boundary layer instability of a natural convection flow on a uniformly heated vertical plate submerged in a homogeneous quiescent environment. A Boussinesq fluid with Prandtl numbers of Pr = 0.733 (air) and 6.7 (water), in the local Rayleigh number range 0 ? Ra_x ? 2.4 × 10¹⁰, is studied. Controlled low amplitude numerical disturbances introduced into the base flow excite unstable travelling waves, with the resulting waves tracked and analyzed as they travel up the boundary layer. The numerical simulation readily reproduced what is predicted by the parallel linear stability theory for the two dimensional mode relatively short wave spectrum, but not for some parts of the long wave spectrum. Critical Rayleigh numbers have been obtained separately for both the temperature and velocity signals using the numerical results, and shown to be in good agreement with each other provided the data is renormalized using the boundary layer scalings of Sparrow and Greg [1]. It has been shown that the disturbance behavior depends on the Prandtl and Rayleigh numbers, the excitation frequency and to a lesser extent the prescribed thermal coupling at the plate.     

Forced convection heat transfer from an elliptical cylinder to power-law fluids

Forced convection heat transfer to incompressible power-law fluids from a heated elliptical cylinder in the steady, laminar cross-flow regime has been studied numerically. In particular, the effects of the power-law index (0.2 ? n ? 1.8), Reynolds number (0.01 ? Re ? 40), Prandtl number (1 ? Pr ? 100) and the aspect ratio of the elliptic cylinder (0.2 ? E ? 5) on the average Nusselt number (Nu) have been studied. The average Nusselt number for an elliptic cylinder shows a dependence on the Reynolds and Prandtl numbers and power-law index, which is qualitatively similar to that for a circular cylinder. Thus, heat transfer is facilitated by the shear-thinning tendency of the fluid, while it is generally impeded in shear-thickening fluids. The average Nusselt number values have also been interpreted in terms of the usual Colburn heat transfer factor (j). The functional dependence of the average Nusselt number on the dimensionless parameters (Re, n, Pr, E) has been presented by empirically fitting the numerical results for their easy use in process design calculations.     

Natural convection and fluid flow in inclined enclosure with a corner heater

The phenomena of natural convection in an inclined square enclosure heated via corner heater have been studied numerically. Finite difference method is used for solving momentum and energy equations in the form of stream function–vorticity. One wall of the enclosure is isothermal but its temperature is colder than that of heaters while the remaining walls are adiabatic. The numerical procedure adopted in this analysis yields consistent performance over a wide range of parameters; Rayleigh number, Ra (10³ ? Ra ? 10⁶); Prandtl number, Pr (0.07 ? Pr ? 70); dimensionless lengths of heater in x and y directions (0.25 ? hx ? 0.75, 0.25 ? hy ? 0.75); and inclination angle, ? (0° ? ? ? 270°). It is observed that heat transfer is maximum or minimum depending on the inclination angle and depending on the length of the corner heaters. The effect of Prandtl number on mean Nusselt number is more significant for Pr < 1.     

Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus

A numerical study for steady-state, laminar natural convection in a horizontal annulus between a heated triangular inner cylinder and cold elliptical outer cylinder was investigated using lattice Boltzmann method. Both inner and outer surfaces are maintained at the constant temperature and air is the working fluid. Study is carried out for Rayleigh numbers ranging from 1.0 × 10³ to 5.0 × 10⁵. The effects of different aspect ratios and elliptical cylinder orientation were studied at different Rayleigh numbers. The local and average Nusselt numbers and percentage of increment heat transfer rate were presented. The average Nusselt number was correlated. The results show that by decreasing the value of aspect ratio and/or increasing the Rayleigh number, the Nusselt number increases. Also the heat transfer rate increases when the ellipse positioned vertically.     

Natural convection in a rectangular enclosure containing an oval-shaped heat source and filled with Fe3O4/water nanofluid

This work concerns with the study of natural convection heat transfer in rectangular cavities with an inside oval-shaped heat source filled with Fe₃O₄/water nanofluid. The finite element method is employed to solve the governing equations for this problem. Average Nusselt numbers are presented for a wide range of Rayleigh number (10³ ≤ Ra ≤ 10⁵), volume fraction of nanoparticles (0 ≤ ϕ ≤ 14%), and four different size and shapes of the heat source. Depending on concentration of the nanoparticle, geometry of the heat source, and the value of Rayleigh number different behaviors are monitored for average Nusselt numbers. Configuration of the heat source dictates a significant change on the behavior of the average Nusselt number, while addition of the nanoparticles has a negative effect on the magnitude of Nusselt number for this problem.     

Jet impingement cooling of a constant heat flux horizontal surface in a confined porous medium: Mixed convection regime

In the present study, numerical investigation of jet impingement cooling of a constant heat flux horizontal surface immersed in a confined porous channel is performed under mixed convection conditions with the limitation of the Darcy model. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 1.0), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. The correlation for Nu_avg in the forced convection regime is suggested. It is shown that mixed convection mode can cause minimum average Nusselt number unfavorably due to counteraction of jet flow against buoyancy driven flow. Hence, careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

Asymptotic analysis of heat transfer in turbulent Rayleigh–Bénard convection

Based on asymptotic considerations a heat transfer law for turbulent Rayleigh–Bénard convection is found that differs from existing correlations which often are of a power law type with respect to their Rayleigh number dependence. From the asymptotic temperature profile, derived by matching temperature gradients in the overlap region of the wall layer and the core layer, a Nusselt number follows which includes a logarithmic term. This correlation is in good agreement with data from highly accurate Rayleigh–Bénard experiments for Rayleigh numbers between 10⁵ and 10¹⁵ and Prandtl numbers larger than 0.5. It is an alternative to existing power laws or more complicated correlations for Nu = Nu(Ra,Pr).     

Effect of aspect ratio on natural convection in power-law liquids from a heated horizontal elliptic cylinder

The influence of aspect ratio and shear-dependent viscosity on free convection heat transfer from a horizontal heated elliptic cylinder in power-law fluids has been investigated. In particular, the coupled momentum and energy equations have been solved numerically over the following ranges of conditions: Grashof number, 10 ? Gr ? 10⁵; Prandtl number, 0.72 ? Pr ? 100; power-law index, 0.3 ? n ? 1.5 and aspect ratio, 0.2 ? E ? 5. The new extensive results demonstrate the influence of the Grashof number (Gr), Prandtl number (Pr), power-law index (n) and aspect ratio (E) on the macroscopic heat and momentum transfer characteristics like local and average values of Nusselt number (Nu) and drag coefficients (C_D). Further insights are developed by examining the structure of the flow and temperature fields adjacent to the cylinder. Broadly speaking, all else being equal, shear- thinning fluid behaviour promotes heat transfer whereas shear-thickening viscosity has a deleterious effect on it with reference to that in Newtonian fluids. Also, the rate of heat transfer gradually increases as the cylinder shape passes from blunt to slender with respect to the direction of gravity. Finally, the present numerical values of the Nusselt number are correlated using a simple analytical form which facilitates interpolation of the present results for the intermediate values of the governing parameters. The paper is concluded by presenting detailed comparisons with the previous numerical and experimental results available in the literature, especially in Newtonian fluids.