MHD natural convection in an enclosure from two semi-circular heaters on the bottom wall

A computational study of the thermal and dynamical behavior of fluid in an enclosure with two isothermal semi-circular heaters is presented. The top wall and the flat surfaces on bottom wall are adiabatic while the vertical walls are kept at lower temperature than the semi-circular heaters. The radius of curvature of the semi-circular surfaces is chosen as one tenth of the cavity wall length. The governing equations are solved by the Galerkin weighted residual finite element method. The effect of magnetic field on the flow is another important parameter in this study. Numerical simulations were performed for several values of Rayleigh number (10³ ? Ra ? 10⁶), Hartmann number (0 ? Ha ? 50) and the distance between two semi-circular heaters (0.2 ? D ? 0.8). In all cases the Prandtl number is taken as 7. It is found that the distance between the semi-circular heaters is the most important parameter affecting the heat and fluid flow fields. In addition, Hartmann number was found to have an adverse affect on heat transfer.     

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.     

Numerical simulations of periodic flow oscillations in low Prandtl number fluids

The transition from steady to oscillatory flow for a very low Prandtl number fluid (Pr = 0.008) is computed for rectangular enclosures with aspect ratios (length/height) of 0.25, 0.4, 1.0, and 2.0 and are found to occur at Rayleigh numbers of 250,000, 130,000, 83,500, and 30,000 respectively. The structures of the oscillations are graphically depicted and are manifested in corner cells which dissipate into centered cells and then into opposite corner cells. A secondary flow transition is detected for a geometry with an aspect ratio of 1.0 at Ra = 1.2Ra_c2.     

Natural convection of water-based nanofluids in an inclined enclosure with a heat source

This study investigates natural convection heat transfer of water-based nanofluids in an inclined square enclosure where the left vertical side is heated with a constant heat flux, the right side is cooled, and the other sides are kept adiabatic. The governing equations are solved using polynomial differential quadrature (PDQ) method. Calculations were performed for inclination angles from 0° to 90°, solid volume fractions ranging from 0% to 20%, constant heat flux heaters of lengths 0.25, 0.50 and 1.0, and a Rayleigh number varying from 10⁴ to 10⁶. The ratio of the nanolayer thickness to the original particle radius is kept at a constant value of 0.1. The heat source is placed at the center of the left wall. Five types of nanoparticles are taken into consideration: Cu, Ag, CuO, Al₂O₃, and TiO₂. The results show that the average heat transfer rate increases significantly as particle volume fraction and Rayleigh number increase. The results also show that the length of the heater is also an important parameter affecting the flow and temperature fields. The average heat transfer decreases with an increase in the length of the heater. As the heater length is increased, the average heat transfer rate starts to decrease for a smaller inclination angle (it starts to decrease with inclination at 90° for ? = 0.25, 60° for ? = 0.50, 45° for ? = 1.0, respectively).     

Laminar natural convection in a square cavity: Low Prandtl numbers and large density differences

Steady natural convection at low Prandtl numbers caused by large density differences in a square cavity heated through the side walls is investigated numerically and theoretically. An appropriate dimensionless parameter characterizing the density differences of the working fluid is identified by the Gay-Lussac number. The Boussinesq assumption is achieved when the Gay-Lussac number tends to zero. The Nusselt number is derived for the ranges in Rayleigh number 10 ? Ra ? 10⁸, in Prandtl number 0.0071 ? Pr ? 7.1 and in Gay-Lussac number 0 ? Ga < 2. The effects of the Rayleigh, Prandtl and Gay-Lussac numbers on the Nusselt number are discussed on physical grounds by means of a scale analysis. Finally, based on physical arguments, a heat transfer correlation is proposed, valid for all Prandtl and Gay-Lussac number ranges addressed.     

Analysis of Bejan’s heatlines on visualization of heat flow and thermal mixing in tilted square cavities

This article analyzes the detailed heat transfer phenomena during natural convection within tilted square cavities with isothermally cooled walls (BC and DA) and hot wall AB is parallel to the insulated wall CD. A penalty finite element analysis with bi-quadratic elements has been used to investigate the results in terms of streamlines, isotherms and heatlines. The present numerical procedure is performed over a wide range of parameters (10³ ? Ra ? 10⁵,0.015 ? Pr ? 1000,0° ? φ ? 90°). Secondary circulations cells are observed near corner regions of cavity for all φ’s at Pr = 0.015 with Ra = 10⁵. Two asymmetric flow circulation cells are found to occupy the entire cavity for φ = 15° at Pr = 0.7 and Pr = 1000 with Ra = 10⁵. Heatlines indicate that the cavity with inclination angle φ = 15° corresponds to large convective heat transfer from the wall AB to wall DA whereas the heat transfer to wall BC is maximum for φ = 75°. Heat transfer rates along the walls are obtained in terms of local and average Nusselt numbers and they are explained based on gradients of heatfunctions. Average Nusselt number distributions show that heat transfer rate along wall DA is larger for lower inclination angle (φ = 15°) whereas maximum heat transfer rate along wall BC occur for higher inclination angle (φ = 75°).     

Investigation of natural circulation in cavities with uniform heat generation for different Prandtl number fluids

Natural convection in enclosures with uniform heat generation and isothermal side walls is studied here. For the rectangular enclosure, two-dimensional conservation equations are solved using SIMPLE algorithm. Parametric studies are conducted to examine the effects of orientation of the cavity, fluid properties (Pr number), and aspect ratio for Rayleigh numbers up to 10⁶. For a horizontal square cavity, the flow becomes periodically oscillating at Ra = 5 × 10⁴ and chaotic at Ra = 8 × 10⁵. With a slight increase in the inclination angle, the oscillations die and for inclination angles greater than 15⁰, the flow attain a steady state over a range of Ra. It is found that for tall cavities (aspect ratio > 1), the steady-state solution is obtained for all values of Ra considered here. However, for wide cavities (aspect ratio < 1), an oscillatory flow regime is observed. The maximum temperature within the cavity is calculated for the range of Ra, aspect ratio and Pr number. Correlations for the maximum cavity temperature is presented here. The values of critical Rayleigh number at which the convection sets in the rectangular cavity are also studied and two distinct criteria are determined to evaluate the critical Rayleigh number. Further, a three-dimensional simulation is performed for a cubic cavity. It is found that the steady state solutions are obtained for all Rayleigh number, except at Ra = 10⁶. This is in contrast to the predictions for a two-dimensional square cavity, which has an oscillatory zone from Ra = 5 × 10⁴ onwards.     

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.     

Thermal lattice-BGK model based on large-eddy simulation of turbulent natural convection due to internal heat generation

To simulate turbulent convection at high Rayleigh number (Ra), we propose a new thermal lattice-BGK (LBGK) model based on large eddy simulation (LES). Two-dimensional numerical simulations of natural convection with internal heat generation in a square cavity were performed at Ra from 10⁶ to 10¹³ with Prandtl numbers (Pr) at 0.25 and 0.60. Simulation results indicate that our model is fit to simulate high Ra flow for its better numerical stability. At Ra = 10¹³, a global turbulent has occurred. With a further increase in Ra, the flow will arrive in a fully turbulence regime. The Nusselt–Rayleigh relationship is also discussed.     

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr = 0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 10³ to 10⁵ and inclination angles γ (0° ? γ < 360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.     

Steady natural convection flow in a square cavity filled with a porous medium for linearly heated side wall(s)

In this paper natural convection flows in a square cavity filled with a porous matrix has been investigated numerically when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Darcy–Forchheimer model without the inertia term is used to simulate the momentum transfer in the porous medium. Penalty finite element method with bi-quadratic rectangular elements is used to solve the non-dimensional governing equations. Numerical results are presented for a range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.2 ⩽ Pr ⩽ 100) in terms of stream functions and isotherm contours, and local and average Nusselt numbers.     

Buoyancy-induced flow and heat transfer in a partially divided square enclosure

The present investigation addressed buoyancy-induced heat transfer in a partially divided square enclosure. The transport equations were solved using the finite element formulation based on the Galerkin method of weighted residuals. The validity of the numerical code used was ascertained by comparing our results with previously published results. Results were obtained in terms of streamlines, isotherms, and Nusselt number for various geometrical parameters specifying the height, width and position of the heater. The effect of Rayleigh number in the range of 10⁴ ? Ra ? 5 × 10⁷ was highlighted in the proposed work. The results revealed that all the parameters related to the geometrical dimensions of the heater were significant on the flow field, isotherms, and heat transfer. The examined dimensionless geometric dimensions employed along with their respective ranges were: heater width (0.005 ? W ? 0.5), heater height (0.005 ? H ? 0.5), and heater location (0 ? D ? 0.5). The investigation revealed that increased heater height, width, and location has enhanced the heat transfer due to increasing the surface area of the heater.     

Heat flow visualization for natural convection in rhombic enclosures due to isothermal and non-isothermal heating at the bottom wall

Analysis has been carried out for the energy distribution and thermal mixing in steady laminar natural convective flow through the rhombic enclosures with various inclination angles, φ for various industrial applications. Simulations are carried out for various regimes of Prandtl (Pr) and Rayleigh (Ra) numbers. Dimensionless streamfunctions and heatfunctions are used to visualize the flow and energy distribution, respectively. Multiple flow circulations are observed at Pr = 0.015 and 0.7 for all φs at Ra = 10⁵. On the other hand, two asymmetric flow circulation cells are found to occupy the entire cavity for φ = 75° at higher Pr (Pr = 7.2 and 1000) and Ra (Ra = 10⁵). Heatlines are found to be parallel circular arcs connecting the cold and hot walls for the conduction dominant heat transfer at Ra = 10³. The enhanced convective heat transfer is explained with dense heatlines and convective loop of heatlines at Ra = 10⁵. Heatlines clearly demonstrate that the left wall receives heat from the bottom wall as heatlines directly connect both the walls whereas the convective heat circulation cells play lead role to distribute the heat along the right wall, especially for smaller φs. On the other hand, the heat flow is evenly distributed to both side walls at higher φs via convection as well as direct conductive transport. Significant convective heat transfer from the bottom hot wall to the left cold wall occurs for φ = 30° cavity whereas the heat transfer to the right cold wall is maximum for φ = 75° irrespective of Pr. Average Nusselt number studies also show that φ = 30° cavity gives maximum heat transfer rate from the bottom to left wall irrespective of Pr in isothermal heating case. On the other hand, enhanced thermal mixing occurs at φ = 75° for both isothermal and non-isothermal heating strategies except at Pr = 0.015 in isothermal heating case.     

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.     

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%.     

Bénard convection from a circular cylinder in a packed bed

Bénard convection around a circular heated cylinder embedded in a packed bed of spheres is studied numerically. The Forchheimer–Brinkman–extended Darcy momentum model with the Local Thermal Non-Equilibrium energy model is used in the mathematical formulation for the porous layer. The governing parameters considered are the Rayleigh number (10³ ≤ Ra ≤ 5 × 10⁷) and the thermal conductivity ratio (0.1 ≤ k_r ≤ 10,000). The structural properties of the packed bed are kept constant as: cylinder-to-particle diameter ratio D/d = 20 and porosity ε = 0.5, while the Prandtl number is fixed at Pr = 0.71. It is found that the presence of the porous medium suppresses significantly the strong free convection produced in the empty enclosure, and reduces considerably the high intensity of the pair of vortices generated behind the cylinder. Also, the results show that the porous medium can play the role of insulator or enhancer of heat transfer from the heat source, depending mainly on their thermal conductivities regardless of the Rayleigh number.     

The temperature statistics of a surfactant-covered air/water interface during mixed convection heat transfer and evaporation

Surface temperature fields were measured of an air/water interface where heat was transferred from the water to the air under mixed convection conditions. The interfacial temperature field was measured using an infrared (IR) camera for mean wind speeds ranging from 0 to 4.0 m/s, in 1.0 m/s increments. Statistics of these surface temperature fields, specifically, the root mean square (rms) and the skewness were obtained. Plots of the rms versus the heat flux showed linear behavior for low wind speeds (U = 0–3 m/s), and the skewness was also found to increase with heat flux for U = 0–3 m/s, although these data exhibited significant scatter. The scaled root mean square temperature was revealed to be governed by the ratio Ra^1/3/(Re^14/5Pr^1/3) where Ra is the Rayleigh number, Re¹ the Reynolds number based on water side friction velocity and Pr is the Prandtl number.     

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.     

Effect of optical properties on oscillatory hydromagnetic double-diffusive convection within semitransparent fluid

The effect of radiative heat transfer on the hydromagnetic double-diffusive convection in two-dimensional rectangular enclosure is studied numerically for fixed Prandtl, Rayleigh, and Lewis numbers, Pr = 13.6, Ra = 10⁵, Le = 2. Uniform temperatures and concentrations are imposed along the vertical walls while the horizontal walls are assumed to be adiabatic and impermeable to mass transfer. The influences of the optical thickness and scattering albedo of the semitransparent fluid on heat and mass transfer with and without magnetic damping are depicted. When progressively varying the optical thickness, multiple solutions are obtained which are steady or oscillatory accordingly to the initial conditions. the mechanisms of the transitions between steady compositionally dominated flow and unsteady thermally dominated flow are analyzed.