Unsteady natural convection within a differentially heated enclosure of sinusoidal corrugated side walls

Numerically investigation of natural convection within a differentially heated modified square enclosure with sinusoidally corrugated side walls has been performed for different values of Rayleigh number. The fluid inside the enclosure considered is air and is quiescent, initially. The top and bottom surfaces are flat and considered as adiabatic. Results reveal three main stages: an initial stage, a transitory or oscillatory stage and a steady stage for the development of natural convection flow inside the corrugated cavity. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. Investigation has been performed for the Rayleigh number, Ra ranging from 10⁵ to 10⁸ with variation of corrugation amplitude and frequency. Constant physical properties for the fluid medium have been assumed except for the density where Boussinesq’s approximation has been considered. Results have been presented in terms of the isotherms, streamlines, temperature plots, average Nusselt numbers, traveling waves and thermal boundary layer thickness plots, temperature and velocity profiles. The effects of sudden differential heating and its consequent transient behavior on fluid flow and heat transfer characteristics have been observed for the range of governing parameters. The present results show that the transient phenomena are greatly influenced by the variation of the Rayleigh number with corrugation amplitude and frequency.     

EFFECT OF TIME MARCHING SCHEMES ON PREDICTIONS OF OSCILLATORY NATURAL CONVECTION IN FLUIDS OF LOW PRANDTL NUMBER

Abstract

A comparative study is presented of two time marching schemes for simulating oscillatory natural convection of low-Prandtl-number fluids in a square cavity. Fully implicit time marching, which is first-order accurate in time, is compared with the second-order-accurate, semi-implicit (Crank-Nicolson) scheme. Both methods are implemented in a control-volume-based finite difference formulation with the central difference scheme used for advection and diffusion terms. Calculations are compared in detail for Gr = 10⁷ and Pr = 0.00S, and results for Gr = 10⁶ and 3 × 10" are briefly summarized. While both time marching schemes predict oscillatory convection at Gr = 10⁷ using an 82 × 82 grid and dimensionless time steps [( Δtα/ L²)Pr√Gr[of 1/80, the flow structure and dynamic behavior predicted by the semi-implicit scheme are more complex than those predicted by the fully implicit scheme. Moreover, it is shown that the semi-implicit calculations are independent of time step while the fully implicit calculations are not.     

Conjugate natural convection heat transfer in an inclined square cavity containing a conducting block

The present work is concerned with computation of natural convection flow in a square enclosure with a centered internal conducting square block both of which are given an inclination angle. Finite volume method through the concepts of staggered grid and SIMPLE algorithm have been applied. Deferred QUICK scheme has been used to discretize the convective fluxes and central difference for diffusive fluxes. The problem of conjugate natural convection has been taken up for validating the code. The abrupt variation in the properties at the solid/fluid interface are taken care of with the harmonic mean formulation. Solution has been performed in the computational domain as a whole with proper treatment at the solid/fluid interface. Computations have been performed for Ra = 10³–10⁶, angle of inclination varying from 15° to 90° in steps of 15° and ratio of solid to fluid thermal conductivities of 0.2 and 5.0. Results are presented in terms of streamlines, isotherms, local and average Nusselt number.     

Heatline method for the visualization of natural convection in a complicated cavity

In this paper, natural convection inside a two-dimensional cavity with a wavy right vertical wall has been carried out. The bottom wall is heated by a spatially varying temperature and other three walls are kept at constant lower temperature. The integral forms of the governing equations are solved numerically using finite-volume method in non-orthogonal body-fitted coordinate system. SIMPLE algorithm with higher-order upwinding scheme are used. The method of numerical visualization of heat transport for convective heat transfer by heatlines is studied. The heatfunction equation in the transformed plane is solved in terms of dimensionless variables. Results are presented in the form of streamlines, isotherms, heatlines, local and average Nusselt number distribution for a selected range of Rayleigh number (10⁰–10⁶). The results are presented for three different undulations (1–3) with different wave amplitude (0.00–0.10) and a fluid having Prandtl number 0.71.     

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.     

Numerical simulation of natural convection heat transfer from a heated square cylinder in a square cavity filled with micropolar fluid

A numerical investigation has been performed to visualize the magnetohydrodynamic natural convective heat transfer from a heated square cylinder situated within a square enclosure subjected to nonuniform temperature distributions on the left wall. The flow inside the enclosure is unsteady, incompressible, and laminar and the working fluid is micropolar fluid with constant Prandtl number (Pr = 7). The governing equations of the flow problem are the conservation of mass, energy, and linear momentum, as well as the angular momentum equations. Governing equations formulated in dimensionless velocity and pressure form has been solved by Marker and Cell method with second-order accuracy finite difference scheme. Comprehensive verification of the utilized numerical method and mathematical model has shown a good agreement with numerical data of other authors. The results are discussed in terms of the distribution of streamlines and isotherms and surface-averaged Nusselt number, for combinations of Rayleigh number, Ra (10³–10⁶), Vortex viscosity parameter, K (0–5), and Ha parameter (0–50). It has been shown that an increase in the vortex viscosity parameter leads to attenuation of the convective flow and heat transfer inside the cavity.     

Conjugate natural convection in an enclosure with a heat source of constant heat transfer rate

A numerical study of two-dimensional transient natural convection in a rectangular enclosure having finite thickness heat-conducting walls with a heat source of constant heat transfer rate located on the inner side of the left wall in conditions of convection–radiation heat exchange with an environment on one of the external boundaries has been performed. Mathematical simulation has been carried out in terms of the dimensionless variables such as stream function – vorticity – temperature. Stream function, vorticity and energy equations have been solved by finite difference numerical method. The relevant governing parameters were: the Grashof number from 10⁶ to 10⁸, the Prandtl number, Pr = 0.7 and the conductivity ratio. Detailed results including streamlines and temperature profiles have been obtained.     

Transient study of buoyancy-assisted mixed convection in laminar plane wall jet flow

A numerical investigation of two-dimensional transient buoyancy-assisted laminar plane wall jet flow has been conducted. The governing equations in the stream function-vorticity formulation have been solved by alternating direction implicit (ADI) method. The parameters used are Grashof number 10⁴ − 10⁷, Prandtl number 0.01, 0.71, 7.1, 10.0 and 15.0 with a constant Reynolds number 400. The plate is considered at a higher temperature and the fluid is assumed to be initially at a uniform cold temperature. The streamline, u-velocity and the isotherm contours are presented at different time levels. The time required for the average Nusselt number to attain a steady-state value decreases with increase in Grashof number.     

Natural convection in enclosures with partial partitions

Laminar and turbulent natural convection in enclosures with partial partitions has been studied by a numerical method. Vertical boundaries were isothermal and horizontal boundaries were adiabatic. Two dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using the Simpler method. Various geometrical parameters were: aspect ratio A=0.3 to 0.4, partition position D=0.5−0.6, height of the partitions C=D=0 to 0.15. The Rayleigh number was varied from 10⁴ to 10¹¹. The results are reduced in terms of the normalized Nusselt number as a function of the Rayleigh number, and other non dimensional geometrical parameters. The isotherms and streamlines are produced for various Rayleigh numbers and geometrical conditions. Heat transfer correlations useful for practical design problems have been derived.     

Effects of corrugation frequency and aspect ratio on natural convection within an enclosure having sinusoidal corrugation over a heated top surface

Natural convection of a two-dimensional laminar steady-state incompressible fluid flow in a modified rectangular enclosure with sinusoidal corrugated top surface has been investigated numerically. The present study has been carried out for different corrugation frequencies on the top surface as well as aspect ratios of the enclosure in order to observe the change in hydrodynamic and thermal behavior with constant corrugation amplitude. A constant flux heat source is flush mounted on the top sinusoidal wall, modeling a wavy sheet shaded room exposed to sunlight. The flat bottom surface is considered as adiabatic, while the both vertical side walls are maintained at the constant ambient temperature. The fluid considered inside the enclosure is air having Prandtl number of 0.71. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. The results in terms of isotherms, streamlines and average Nusselt numbers are obtained for the Rayleigh number ranging from 10³ to 10⁶ with constant physical properties for the fluid medium considered. It is found that the convective phenomena are greatly influenced by the presence of the corrugation and variation of aspect ratios.     

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

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

Parametric study on mixed convection heat transfer in an inclined arc-shape cavity

This paper presents a parametric study on mixed convection heat transfer in an inclined arc-shape cavity subjected to a moving lid. The governing equations for the inclined arc-shape cavity were derived with the incorporation of inertia and buoyant force terms and solved by using the finite-volume method and numerical grid generation scheme. The parametric study considered three physical parameters including inclination angle, Reynolds number and Grashof number, and explored the effect of these parameters on the flow field and heat transfer characteristics. Computations were conducted for the Reynolds number ranging from 100 to 1500, Grashof number from 10⁵ to 10⁷ and inclination angle from 15⁰ to 60⁰. The numerical results show that the flow pattern becomes inertia-dominant and the strength of the primary vortex generally increases as the Reynlods number increases. As the Grashof number increases, the strength of the inertial-induced vortex decreases and the strength of the buoyancy-induced vortex increases. The strength of the vortexes decreases with the increasing inclination angle and the buoyancy-induced flow becomes more dominant. The average Nusselt number increases as the Grashof number increases for all the inclination angles studied here. The local friction increases with the increasing inclination angle, and becomes significant as the Grashof number increases.     

CFD-based analysis of entropy generation due to mean and fluctuating fields during turbulent natural convection in a square enclosure

The current study aims to numerically investigate the entropy generation during the natural convection flow of air in a square cavity. The governing equations for the conservation of mass, momentum, energy, and turbulence are solved using a control volume-based technique employing the commercial code Fluent. Runs have been performed for both laminar and turbulent flow regimes by varying the Rayleigh number (Ra) from 10³ to 10¹⁰. On the other hand, various viscous distribution coefficients (ϕ = 10⁻⁴, 10⁻³, and 10⁻²) and constant Prandtl number (Pr = 0.71) were considered. Given the conflicting perspectives in the literature regarding the entropy generation under turbulent regimes, more research is needed to better understand the impact that the fluctuating flow has on entropy production. The four terms of entropy generation inherent to turbulent natural convection (entropy generation due to dissipation in the mean and the fluctuating velocity fields in addition to the heat flux due to the mean and the fluctuating temperature) are computed in the present work and compared to calculations based on only mean values of temperature and velocity gradients. It was found that taking into account the fluctuating terms of temperatures and velocities augment the total entropy generation by 10.10%, 14.43%, and, 17.70%, up to 32.60%, respectively, for Ra = 5 × 10⁸, Ra = 10⁹, Ra = 1.58 × 10⁹, and Ra = 10¹⁰. The gain shows the tendency to increase with the Rayleigh number. Thus, the fluctuating terms cannot be neglected particularly for high Rayleigh numbers. Furthermore, unlike entropy production due to the mean flow field, numerical outcomes reveal that the generated irreversibilities due to fluctuating flow are located around the upper hot and the lower cold corners of the heated walls. In addition, a numerical relationship between the first and the second laws of thermodynamics has been derived. A promising result that emerged from this study has shown that the Nusselt number and therefore the first law of thermodynamics is sufficient to estimate the heat part of entropy generation without the necessity of using the second law.     

Visualization of Heat Transport during Natural Convection Within Porous Triangular Cavities via Heatline Approach

In this article, natural convection in a porous triangular cavity has been analyzed. Bejan's heatlines concept has been used for visualization of heat transfer. Penalty finite-element method with biquadratic elements is used to solve the nondimensional governing equations for the triangular cavity involving hot inclined walls and cold top wall. The numerical solutions are studied in terms of isotherms, streamlines, heatlines, and local and average Nusselt numbers for a wide range of parameters Da (10^?5–10^?3), Pr (0.015–1000), and Ra (Ra = 10³–5 × 10⁵). For low Darcy number (Da = 10^?5), the heat transfer occurs due to conduction as the heatlines are smooth and orthogonal to the isotherms. As the Rayleigh number increases, conduction dominant mode changes into convection dominant mode for Da = 10^?3, and the critical Rayleigh number corresponding to the on-set of convection is obtained. Distribution of heatlines illustrate that most of the heat transport for a low Darcy number (Da = 10^?5) occurs from the top region of hot inclined walls to the cold top wall, whereas heat transfer is more from the bottom region of hot inclined walls to the cold top wall for a high Darcy number (Da = 10^?3). Interesting features of streamlines and heatlines are discussed for lower and higher Prandtl numbers. Heat transfer analysis is obtained in terms of local and average Nusselt numbers (Nu _l , Nu _t ) and the local and average Nusselt numbers are found to be correlated with heatline patterns within the cavity.     

TRANSIENT LAMINAR NATURAL CONVECTION IN AN ENCLOSURE PARTITIONED BY AN ADIABATIC BAFFLE

Transient laminar natural convection in a two-dimensional enclosure partitioned by an adiabatic baffle is investigated numerically, A penalty finite-element method with a Newton-Raphson iteration algorithm and a backward difference scheme dealing with the time term are adopted to solve governing equations. The effects of the baffle and Rayleigh number on the heat transfer mechanism are found to be substantial during the transient process for Rayleigh numbers of 10⁴ and 10⁶. However, the variations of the heat transfer mechanism occur mainly in the first one-third of the time period of the transient, in spite of the presence, absence, or location of a baffle.     

Optical and electrochemical studies of transient free convection mass transfer at horizontal surfaces

The onset of free convection at upward facing horizontal surfaces adjacent to a free fluid environment has been studied using the electrochemical system involving the electrodeposition of Cu²⁺ ions. A transient potential step technique was employed and photography of the ensuing developing convective structure was synchronised with the recording of current-time transients. This enabled the interdependence of instantaneous mass-transfer rate and convective flow structure to be visualised. A number of distinctive flow structures in the transient and steady states were observed and these have been related to the correlation regimes for the overall mass-transfer rate. Interesting phenomena involving periodic instabilities have been discovered at transitional Rayleigh numbers leading to pulsations in the convective plume and corresponding regular oscillations in the steady state mass-transfer rate.The effect of surface roughening on mass transfer and hydrodynamic performance during extended experiments has been observed and a method of overcoming surface roughening difficulties in the application of the electrochemical technique is suggested.Diffusion transient undershoot times are correlated in terms of plate Rayleigh number by the equations in the range 10⁷ < Re_d < 10¹¹ and in the range 10⁴ < Ra_d < 10⁷.The periodic fluctuation frequencies are similarly correlated by the equation .     

Conjugate turbulent natural convection with surface radiation in air filled rectangular enclosures

Conjugate turbulent natural convection and surface radiation in rectangular enclosures heated from below and cooled from other walls, typically encountered in Liquid Metal Fast Breeder Reactor (LMFBR) subsystems, have been investigated by a finite volume method for various aspect ratios. The formulation comprises the standard two equation k–ε turbulence model with physical boundary conditions (no wall functions), along with the Boussinesq approximation, for the flow and heat transfer. As far as radiation is concerned, the radiosity – irradiation formulation for a transparent fluid of Prandtl number 0.7 has been employed. The conjugate coupling on the walls has been handled by using a fin type formulation. The Rayleigh number based on the width of the enclosure is varied from 10⁸ to 10¹² and the aspect ratio is varied from 0.5 to 2.0. Detailed results including stream lines, temperature profiles, and convective, radiative and overall Nusselt numbers are presented. A correlation for the mean convection Nusselt number in terms of Rayleigh number and aspect ratio is proposed for design purposes. The influence of the wall emissivity and the external heat transfer coefficient on the heat transfer from the enclosure has also been investigated.     

Numerical Analysis of the Effects of Selected Geometrical Parameters and Fluid Properties on MHD Natural Convection Flow in an Inclined Elliptic Porous Enclosure with Localized Heating

Magnetohydrodynamic (MHD) natural convection flow and associated heat convection in an oriented elliptic enclosure has been investigated with numerical simulations. A magnetic field was applied to the cylindrical wall of the configuration, the top and bottom walls of the enclosure were circumferentially cooled and heated, respectively, while the extreme ends along the cross‐section of the elliptic duct were considered adiabatic. The full governing equations in terms of continuity, momentum, and energy transport were transformed into nondimensional form and solved numerically using finite difference method adopting Gauss–Seidel iteration technique. The selected geometrical parameters and flow properties considered for the study were eccentricity (0, 0.2, 0.4, 0.6, and 0.8), angle of inclination (0°, 30°, 60°, and 90°), Hartmann number (0, 25, and 50), Grashof number (10⁴, 10⁵, and 10⁶), and Darcy number (10^?3, 10^?4, and 10^?5). The Prandtl number was held constant at 0.7. Numerical results were presented by velocity distributions as well as heat transfer characteristics in terms of local and average Nusselt numbers (i.e., rate of heat transfer). The optimum heat transfer rate was attained at e value of 0.8. Also, the heat transfer rate increased significantly between the angles of inclination 58° and 90°. In addition, Hartmann number increased with decreased heat transfer rate and flow circulation. A strong flow circulation (in terms of velocity distribution) was observed with increased Grashof and Darcy numbers. The combination of the geometric and fluid properties therefore can be used to regulate the circulation and heat transfer characteristics of the flow in the enclosure.     

Numerical simulation of turbulent natural convection in a rectangular enclosure having finite thickness walls

Turbulent natural convection in a rectangular enclosure having finite thickness heat-conducting walls at local heating at the bottom of the cavity provided that convective-radiative heat exchange with an environment on one of the external borders has been numerically studied. Mathematical simulation has been carried out in terms of the dimensionless Reynolds averaged Navier–Stokes (RANS) equations in stream function–vorticity formulations. The formulation comprises the standard two equation k–ε turbulence model with wall functions, along with the Boussinesq approximation, for the flow and heat transfer. The special attention was paid to the effects of the Grashof number 10⁸ ? Gr < 10¹⁰, the transient factor 0 < τ < 1000 and the thermal conductivity ratio k_2,1 = 5.7 × 10^?4, 6.8 × 10^?5 both on local and on integral problem parameters. Detailed results including stream lines, temperature profiles and correlation for the average Nusselt number in terms of Grashof number have been obtained.     

Simulation of mixed convection heat transfer in a horizontal channel with an open cavity containing a heated hollow cylinder

The objective of this paper is to numerically investigate the mixed convective flow and heat transfer controlled by a heated hollow cylinder inside an open cavity attached with a horizontal channel. All the boundaries of the channel and cavity are perfectly insulated while the inner surface of the cylinder is heated uniformly by heat flux q. The equations of conservation of mass, momentum, and energy were solved using adequate boundary conditions by Galarkin's weighted residual finite element technique. The solution has been performed in the computational domain as a whole with proper treatment at the solid/fluid interface. Computations have been conducted for Ra = 10³–10⁵, Prandtl number Pr varying from 0.7 to 7 and ratio of solid to fluid thermal conductivities from 0.2 to 50. Results are presented in terms of streamlines, isotherms, heat transfer rate in terms of the average Nusselt number (Nu_av), drag force (D), and maximum bulk temperature (θ_max). © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21002