MHD natural convection in a laterally and volumetrically heated square cavity

A numerical study is presented of unsteady two-dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. The flow is characterized by the external Rayleigh number, Ra_E, determined from the temperature difference of the side walls, the internal Rayleigh number, Ra_I, determined from the volumetric heat rate, and the Hartmann number, Ha, determined from the strength of the imposed magnetic field. Starting from given values of Ra_E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra_I/Ra_E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multicellular flow up to a threshold value, Ra_I,cr, of the internal Rayleigh number depending on Ha. Oscillatory multicellular flow fields were observed for S values up to 100 for the range 10⁵-10⁶ of Ra_E studied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Hartmann number is not found to be straightforward connected with the resulting flow patterns in the present flow configuration.     

Natural convection in a partially open square cavity with internal heat source: An analysis of the opening mass flow

A steady buoyancy-driven flow of air in a partially open square 2D cavity with internal heat source, adiabatic bottom and top walls, and vertical walls maintained at different constant temperatures is investigated numerically in this work. A heat source with 1% of the cavity volume is present in the center of the bottom wall. The cold right wall contains a partial opening occupying 25%, 50% or 75% of the wall. The influence of the temperature gradient between the verticals walls was analyzed for Ra_e = 10³–10⁵, while the influence of the heat source was evaluated through the relation R = Ra_i/Ra_e, investigated at between 400 and 2000. Interesting results were obtained. For a low Rayleigh number, it is found that the isotherm plots are smooth and follow a parabolic shape indicating the dominance of the heat source. But as the Ra_e increases, the flow slowly becomes dominated by the temperature difference between the walls. It is also observed that multiple strong secondary circulations are formed for fluids with a small Ra_e whereas these features are absent at higher Ra_e. The comprehensive analysis is concluded with horizontal air velocity and temperature plots for the opening. The numerical results show a significant influence of the opening on the heat transfer in the cavity.     

Conjugate natural convection in a square cavity with finite thickness horizontal walls

The effect of conduction of horizontal walls on natural convection heat transfer in a square cavity is numerically investigated. The vertical walls of the cavity are at different constant temperatures while the outer surfaces of horizontal walls are insulated. A code based on vorticity–stream function is written to solve the governing equations simultaneously over the entire computational domain. The dimensionless wall thickness of cavity is taken as 0.1. The steady state results are obtained for wide ranges of Rayleigh number (10³ < Ra < 10⁶) and thermal conductivity ratio (0 < K < 50). The variation of heat transfer rate through the cavity and horizontal walls with Rayleigh number and conductivity ratio is analyzed. It is found that although the horizontal walls do not directly reduce temperature difference between the vertical walls of cavity, they decrease heat transfer rate across the cavity particularly for high values of Rayleigh number and thermal conductivity ratio. Heatline visualization technique is a useful application for conjugate heat transfer problems as shown in this study.     

Conjugate natural convection in air filled tube inserted a square cavity

Numerical analyses of fluid flow and heat transfer due to buoyancy forces in a tube inserted square cavity filled with fluid were carried out by using control volume method in this study. The cavity was heated from the left wall and cooled from the right isothermally and horizontal walls were adiabatic. A circular tube filled with air was inserted into the square cavity. The case that the inside and outside of the tube were filled with the same fluid (air) was examined. Varied solid materials were chosen as the tube wall. Results were obtained for different Rayleigh numbers (Ra = 10⁴, 10⁵ and 10⁶), thermal conductivity ratio of the fluid to the tube wall (k = 0.1, 1 and 10) and different location centers of the tube (c (0.25 ≤ x ≤ 0.75, 0.25 ≤ y ≤ 0.75)). Comparison with benchmark solutions of the natural convection in a cavity was performed and numerical results gave an acceptable agreement. It was found that varied location of the tube center can lead to different flow fields and heat transfer intensities which are also affected by the value of Rayleigh number.     

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ? = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra_I,cr^?, the convection transits from unicellular to bicellular mode, as the annulus T_max becomes higher than the fixed hot-wall temperature. The Ra_I,cr^? increases proportionately when the permeability based external Rayleigh number Ra_E^? and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra_E^?, Ra_I^?, Da and γ. It predicts the results within ± 20% accuracy.     

Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls

In the present study, entropy generation in rectangular cavities with the same area but different aspect ratios is numerically investigated. The vertical walls of the cavities are at different constant temperatures while the horizontal walls are adiabatic. Heat transfer between vertical walls occurs by laminar natural convection. Based on the obtained dimensionless velocity and temperature values, the distributions of local entropy generation due to heat transfer and fluid friction, the local Bejan number and local entropy generation number are determined and related maps are plotted. The variation of the total entropy generation and average Bejan number for the whole cavity volume at different aspect ratios for different values of the Rayleigh number and irreversibility distribution ratio are also evaluated. It is found that for a cavity with high value of Rayleigh number (i.e., Ra = 10⁵), the total entropy generation due to fluid friction and total entropy generation number increase with increasing aspect ratio, attain a maximum and then decrease. The present results are compared with reported solutions and excellent agreement is observed. The study is performed for 10² < Ra < 10⁵, 10^− 4 < ? < 10^− 2, and Pr = 0.7.     

Laminar and turbulent free convection in a composite enclosure

Turbulent natural convection in a two-dimensional horizontal composite square cavity, isothermally heated at the left side and cooled from the opposing surface, is numerically analyzed using the finite volume method. The composite square cavity is formed by three distinct regions, namely, clear, porous and solid region. The development of a numerical tool able to treat all these regions as one computational domain is of advantage for engineering design and analysis of passive thermal control systems. Governing equations are written in terms of primitive variables and are recast into a general form. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 10⁶. Nusselt number values show that for the range of Ra analyzed there is no significant variation between the laminar and turbulent model solution. When comparing the effects of Ra, k_s/k_f and Da on Nu, results indicate that the solid phase properties have a greater influence in enhancing the overall heat transferred trough the cavity.     

Investigation of buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water–Cu nanofluid

A numerical study is conducted to investigate the transport mechanism of free convection in a trapezoidal enclosure filled with water–Cu nanofluid. The horizontal walls of the enclosure are insulated while the inclined walls are kept at constant but different temperatures. The numerical approach is based on the finite element technique with Galerkin's weighted residual simulation. Solutions are obtained for a wide range of the aspect ratio (AR) and Prandtl number (Pr) with Rayleigh number (Ra = 10⁵) and solid volume fraction (? = 0.05). The streamlines, isotherm plots and the variation of the average Nusselt number at the left hot wall are presented and discussed. It is found that both AR and Pr affect the fluid flow and heat transfer in the enclosure. A correlation is also developed graphically for the average Nusselt number as a function of the Prandtl number as well as the cavity aspect ratio.     

Coupled Natural Convection and Radiation in a Horizontal Rectangular Enclosure Discretely Heated from Below

A numerical study is carried out to investigate the interaction between natural convection and thermal radiation in a horizontal enclosure filled with air and heated discretely from below. The results are presented for a cavity having an aspect ratio A _r  = L′/H′ = 10, while the Rayleigh number and the emissivity of the walls are varied in the ranges 10³ ≤ Ra ≤ 10⁶ and 0 ≤ ε ≤ 1, respectively. The results of the study, presented in terms of flow and temperature patterns, average convective, radiative and total Nusselt numbers, evaluated on the cold wall, show that the problem has multiple solutions. Each of these solutions is characterized by a specific flow structure, and its appearance and range of existence depend strongly on the parameters Ra and ε. The amount of heat evacuated through the cold surface is dependent on the type of solution.     

Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model

This work presents numerical computations for laminar and turbulent natural convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < Ra_m < 500 and 3.2 × 10⁻⁴ > Da > 3.2 × 10⁻⁶ and made use of the finite volume method. The inner and outer walls are maintained at constant but different temperatures. The macroscopic k–ε turbulence model with wall function is used to handle turbulent flows in porous media. First, the turbulence model is switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ra_m. Subsequently, the turbulence model is included and calculations start at high Ra_m, merging to the laminar branch for a reducing Ra_m. This convergence of results as Ra_m decreases can be seen as an estimate of the so-called laminarization phenomenon. Here, a critical Rayleigh number was not identified and results indicated that when the porosity, Prandtl number, conductivity ratio between the fluid and the solid matrix and Ra_m are kept fixed, the lower the Darcy number, the higher is the difference of the average Nusselt number given by the laminar and turbulent models.     

Influence of surface radiation on the transition to unsteadiness for a natural convection flow in a differentially heated cavity

Abstract

The influence of surface radiation on the transition to the unsteady state in natural convection is studied numerically. The configuration of the differentially heated square cavity with adiabatic horizontal walls is chosen to generate an internal natural convection flow. It is known that radiative transfers reduce the temperature difference between the adiabatic walls, which consequently reduces the thermal stratification of the central zone and increases the velocity flow. Many studies have focused on the stationary regime, but few of them have investigated the transition to unsteady flow. For this purpose, the effect of the wall emissivity on the critical Rayleigh number and the associated critical frequency was studied for a given cavity length. The cavity length and mean temperature of isothermal walls are set for the whole study. The results show that all these values are between the values obtained without radiation and those obtained for perfectly conducting horizontal walls. The critical Rayleigh number decreases with emissivity while the associated frequency increases. Moreover, the symmetry of fluctuating properties of the flow is changed when the radiation is taken into account.     

Numerical study of mixed convection in a ventilated square enclosure with the lattice Boltzmann method

In this article, numerical study of heat transfer by convection in a square cavity was investigated. The vertical walls of the cavity are differentially heated and the horizontal walls are considered adiabatic. A ventilation jet is created by a fan placed in the cavity. A lattice Boltzmann model for incompressible flow equation is used to simulate the problem. A parametric study was performed presenting the influence of Reynolds number (20 ≤ Re?≤?500), Rayleigh number (10≤Ra?≤?10⁺⁶), and fan position (0.2?≤?L_F≤0.8). It has been observed that heat transfer rate increases with the Reynolds number increasing and it is maximal for the L_F=0.2.     

Magnetohydrodynamic mixed convection in a horizontal channel with an open cavity

The development of magnetic field effect on mixed convective flow in a horizontal channel with a bottom heated open enclosure has been numerically studied. The enclosure considered has rectangular horizontal lower surface and vertical side surfaces. The lower surface is at a uniform temperature T_h while other sides of the cavity along with the channel walls are adiabatic. The governing two-dimensional flow equations have been solved by using Galarkin weighted residual finite element technique. The investigations are conducted for different values of Rayleigh number (Ra), Reynolds number (Re) and Hartmann number (Ha). Various characteristics such as streamlines, isotherms and heat transfer rate in terms of the average Nusselt number (Nu), the Drag force (D) and average bulk temperature (θ_av) are presented. The results indicate that the mentioned parameters strongly affect the flow phenomenon and temperature field inside the cavity whereas in the channel these effects are less significant.     

Magnetic field effect on the unsteady free convection flow in a square cavity filled with a porous medium with a constant heat generation

The effects of an inclined magnetic field and heat generation on unsteady free convection within a square cavity filled with a fluid-saturated porous medium have been investigated numerically. The top and bottom horizontal walls of the enclosure are adiabatic whereas the vertical walls are kept at constant but different temperatures. The physical problems are represented mathematically by a set of partial differential equations along with the corresponding boundary conditions. By using an implicit finite-difference scheme, namely the ADI method (Alternative Direction Implicit), the non-dimensional governing equations are numerically solved. The influential parameters are the Rayleigh number Ra, the inclination angle γ of the magnetic field relative to the gravity vector g, the Hartmann number Ha and the heat generation parameter Q. In the present study, the obtained results are presented in terms of streamlines, isotherms and average Nusselt number along the hot wall. The result shows that with increasing Ha, the diffusive heat transfer become prominent even though the Rayleigh number increases.     

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.     

NATURAL CONVECTION IN A TWO-DIMENSIONAL DIFFERENTIALLY HEATED SQUARE ENCLOSURE UNDERGOING ROTATION

A numerical study of two-dimensional natural convection in a rotating and differentially heated square enclosure has been presented by solving the conservation equations of mass, momentum, and energy in a rotating coordinate system using the finite difference method. Considering air to be the fluid medium in the cavity, the results are presented for a wide range of Rayleigh numbers (Ra), Taylor numbers (Ta), and rotational Rayleigh numbers (Ra _w). It is found that a significant enhancement in heat transfer can be achieved due to rotational effects. At a particular Ra, increase in Ta results in an increase in frequency of oscillations of the dynamical variables and also gives rise to formation of a mushroom-shaped plume in the core of the cavity. For constant Ta, an increase in Ra results in formation of thinner thermal boundary layers at the isothermal walls and stable thermal stratification in the core of the cavity. The stratification becomes unstable when the Ta     

NATURAL CONVECTION IN A DIFFERENTIALLY HEATED SQUARE CAVITY WITH INTERNAL HEAT GENERATION

A high-resolution, finite-difference numerical study is reported on natural convection in a square cavity. The vertical sidewatts of the cavity are differentially heated, and a uniform internal heat generation is also present. Two principal parameters are considered, the internal Rayleigh number Ra_I, which represents the strength of the internal heat generation, and the external Rayleigh number Rag, which denotes the effect due to the differential heating of the side walls. The internal Rayleigh number varies in the range 10¹⁰ Ra_I ≤ 10⁷, while the external Rayleigh number is set at Ra_E = 5 x 10⁷ for most computations. As the relative strength of the internal heat generation increases, the flows near the tap portion of the heated sidewall are directed downward. When the effect of the internal heat generation is dominant, the thermal energy leaves the system for the surroundings over the top portion of the heated wall. Only in the bottom pari of the heated wall is heat transfer directed into the system. These numerical solutions are in qualitative agreement with the available experimental measurements.     

Heat transfer enhancement in cubical enclosures with vertical fins

Natural convection of air in a cubical enclosure with a thick partition fitted vertically on the hot wall is numerically investigated for Rayleigh numbers of 10³–10⁶. A three dimensional convective circulation is generated, in which the cold flow sweeps the fin faces and the hot wall, with low flow blockage. The combined contributions of these faces cause heat transfer enhancements over 40% at high Rayleigh numbers and thermal conductivity ratios (R_k). These enhancements significantly exceed the ones obtained with horizontal fins. Even low R_k values cause heat transfer enhancements, except at Ra = 10⁴.     

Computation of turbulent free convection in left and right tilted porous enclosures using a macroscopic k–ε model

Comparisons of computations for turbulent natural convection within clockwise and counter-clockwise inclined cavities, filled with a fluid saturated porous medium, are presented. The finite volume method in a generalized coordinate system is applied. Oblique walls are maintained at constant but different temperatures, whereas horizontal surfaces are kept insulated. Flow and heat transfer characteristics are investigated for Rayleigh number up to 10⁴ and inclination angles up to 45°, in both directions of rotation. Turbulent is handled using a macroscopic two-equation model with a wall function. In this work, the turbulence model is first switched off and the laminar branch of the solution is obtained. Subsequently, the turbulence model is included and the solution merges to the laminar branch for a reducing value of Ra_m. Present computations are compared with published results and the influence of the inclination angle on Ra_cr is analyzed, for both the left and right rotating directions. For Ra_m greater than around 10⁴, both laminar and turbulent flow solutions deviate, possibly indicating that a critical value for Ra_m was reached. Both left and right rotation of the hot wall reduce Nu, but rotating the hot wall on the counter-clockwise direction decreases Nu at a faster rate than when bending the cavity to the right.