Transient natural convection in a rectangular enclosure with one heated side wall

This paper describes a numerical and theoretical study of the transient natural convection heating of a two-dimensional rectangular enclosure filled with fluid. The heating is applied suddenly along one of the side walls, while the remaining three walls are maintained insulated. It is shown that the process has two distinct phases, an early period dominated by conduction and a late period dominated by convection. The scaling laws for the heat transfer rate and the effectiveness (energy storage fraction) are determined based on scale analysis. These theoretical results are confirmed by numerical experiments conducted in the domain Ra = 10³−10⁶, Pr = 7, A = 1, where Ra is the Rayleigh number based on height and initial temperature difference, Pr is the Prandtl number, and A is the height/length ratio of the enclosure. Correlations for heat transfer rate and effectiveness are constructed by comparing the theoretical scaling laws with the numerical results.     

Transient natural convection of low-Prandtl-number fluids in a differentially heated cavity

The transient convective motion in a two-dimensional square cavity driven by a temperature gradient is analysed. The cavity is filled with a low-Prandtl-number fluid and the vertical walls are maintained at constant but different temperatures, while the horizontal boundaries are adiabatic. A control volume approach with a staggered grid is employed to formulate the finite difference equations. Numerically accurate solutions are obtained for Prandtl numbers of 0·001, 0·005 and 0·01 and for Grashof numbers up to 1 × 10⁷. It was found that the flow field exhibits periodic oscillation at the critical Grashof numbers, which are dependent on the Prandtl number. As the Prandtl number is decreased, the critical Grashof number and the frequency of oscillation decrease. Prior to the oscillatory flow, steady state solutions with an oscillatory transient period were predicted. In addition to the main circulation, four weak circulations were predicted at the corners of the cavity.     

Three-dimensional numerical simulation of periodic natural convection in a differentially heated cubical enclosure

A high-resolution, finite-difference numerical study is carried out of three-dimensional unsteady periodic natural convection of air in a cubical enclosure at the Rayleigh number of 8.5×10⁶. The enclosure is subjected to differential heating at the two vertical side walls. The other vertical walls are insulated. A linear temperature profile is specified at the thermally-conducting horizontal walls. Flow details in the three-dimensional field are captured by elaborate post-processing of the computational results, for which the state-of-the-art numerical visualization techniques are utilized. The three-dimensionality of the mean flow fields is observed to be confined into narrow regions near the end walls. The time-dependent solutions clearly indicate the periodic nature of the flow. The oscillation frequency is in close agreement with the previous experimental measurements reported in the literature.Toru Fusegi is presently at Heat Transfer and Fluid Dynamics, Energy Technology Research Institute, Tokyo Gas Co. Ltd., 1-16-25 Shibaura, Minato, Tokyo 105, Japan.     

Double-diffusive natural convection in a porous enclosure partially heated from below and differentially salted

This paper reports numerical results of two-dimensional double-diffusive natural convection in a square porous cavity partially heated from below while its upper surface is cooled at a constant temperature. The vertical walls of the porous matrix are subjected to a horizontal concentration gradient. The parameters governing the problem are the thermal Rayleigh number (Ra=100 and 200), the Lewis number (Le=0.1, 1 and 10), the buoyancy ratio (−10N10) and the relative position of the heating element with respect to the vertical centerline of the cavity (δ=0 and 0.5). The effect of the governing parameters on fluid characteristics is analyzed. The multiplicity of solutions is explored and the existence of asymmetric bicellular flow is proved when the heated element is shifted towards a vertical boundary (δ=0.5). The solutal buoyancy forces induced by horizontal concentration gradient lead to the elimination of the multiplicity of solutions obtained in pure thermal convection when N reaches some threshold value which depends on Le and Ra.     

Laminar forced convection of power-law non-Newtonian fluids inside ducts

Thermal entrance region heat transfer for laminar forced convection of power-law fluids inside a circular tube and parallel plate channel for uniform wall temperature is solved exactly, and as many eigenvalues and eigenfunctions as needed for the solution are determined automatically and with high accuracy by using the recently advanced Sign-Count method. Results are presented for the local and average Nusselt number over a wide range of the Graetz number in both graphical and tabular forms. The present benchmark results are utilized to critically examine the accuracy of the approximate Leveque solution.

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Laminare Zwangskonvektion von nicht-Newtonschen Fluiden, die in Kanälen strömen und dem Potenzansatz folgen

Zusammenfassung Es werden exakte Lösungen für den Wärmetransport in der thermischen Einlaufzone in runden Rohren und zwischen parallelen Platten für Fluide nach dem Exponentialansatz bei laminarer Zwangskonvektion und mit gleichmäßiger Wandtemperatur angegeben. Unter Benutzung der jüngst verbesserten Sign-Count-Methode werden so viele Eigenwerte und Eigenfunktionen, wie für die Lösung benötigt, automatisch und mit großer Genauigkeit bestimmt. Ergebnisse werden in graphischer wie tabellarischer Form über einen weiten Bereich der Graetz-Zahl für die örtliche und mittlere Nusselt-Zahl vorgestellt. Die vorliegenden richtungsweisenden Ergebnisse werden dazu benützt, um die Genauigkeit der Levequeschen Näherungslösung kritisch zu prüfen.

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Nomenclature b radius of circular duct or half the spacing between parallel plates - C - D _h hydraulic diameter=4b for parallel plate, 2b for circular tube - f(r), F(r) temperature distribution at the inlet, dimensional and dimensionless, respectively - g (r), G (R) energy generation, dimensional and dimensionless, respectively - h (z) heat transfer coefficient - H _i(Z) defined by Eq. (11b) - k thermal conductivity - l₀,l _N reference lengths to nondimensionalizer andz coordinates respectively (l ₀=b andl _N= D_h) - n power-law index - Nu _av average Nusselt number=h _av D_h/k - Nu (Z) local Nusselt number=h (z)D _h/k - p 0 for parallel-plate duct, 1 for circular duct - r radial or normal coordinate - R = dimensionless radial coordinate - T(r, z) fluid temperature - T _av (z) average fluid temperature - T ^* reference temperature - T reference temperature difference - U(R) = normalized velocity profile - U ^*(R) - w (r) fully developed velocity profile - w _av average velocity - z axial coordinate - Z = dimensionless axial coordinate Greek letters thermal diffusivity - (R,Z) = dimensionless temperature - _w = dimensionless wall temperature - _i eigenvalues of the eigenvalue problem (10) - (_i,R) eigenfunctions of the eigenvalue problem (10)     

Natural convection flow under a magnetic field in an inclined square enclosure differentialy heated on adjacent walls

Steady, laminar, natural-convection flow in the presence of a magnetic field in an inclined square enclosure differentially heated along the bottom and left vertical walls while the other walls are kept isothermal was considered. The governing equations were solved numerically for the stream function, vorticity and temperature ratio using the differential quadrature method for various Grashof and Hartmann numbers, inclination angle of the enclosure and direction of the magnetic field. The orientation of the enclosure changes the temperature gradient inside and has a significant effect on the flow pattern. Magnetic field suppresses the convective flow and its direction also influences the flow pattern, causing the appearance of inner loops and multiple eddies. The surface heat flux along the bottom wall is slightly increased by clockwise inclination and reduced by half by the counterclockwise inclination. The surface heat flux along the upper portion of the left side wall is reversed by the rise of warmer fluids due to the convection currents for no inclination and clockwise inclination of the enclosure.     

Interaction of mixed convection in two-sided lid driven differentially heated square enclosure with radiation in presence of participating medium

The current study addresses the mathematical modeling aspects of transport phenomena in steady, two-dimensional, laminar flow accompanied by heat transfer in a lid-driven differentially heated cavity in presence of radiatively absorbing, emitting and scattering gray medium. The walls of the enclosure are considered to be opaque, diffusive and gray. Mixed convection is the outcome of the interaction of forced convection induced by the moving vertical hot and cold wall with the natural convection induced due to the differentially heated enclosure. Two different orientations of the wall movement have been considered to simulate opposing and aiding mixed convection phenomenon and to study its interaction with radiation. Vorticity-stream function formulation of N–S equation has been employed. The discrete ordinate method has been used in modeling the radiative transport equation followed with finite volume method as discretisation technique. The effect of influencing parameters on fluid flow and heat transfer has been studied.     

Unsteady natural convection in an enclosure with vertical wavy walls

In this paper, unsteady heat transfer and fluid flow characteristics in an enclosure are investigated. The enclosure consists of two vertical wavy and two horizontal straight walls. The top and the bottom walls are considered adiabatic. Two wavy walls are kept isothermal and their boundaries are approximated by a cosine function. Governing equations including continuity, momentum and energy were discretized using the finite-volume method and solved by SIMPLE method in curvilinear coordinate. Simulation was carried out for a range of Grashof number Gr = 10³–10⁶, Prandtl number Pr = 0.5–4.0, wave ratio A (defined by amplitude/wavelength) 0.0–0.35 and aspect ratio W (defined by average width/wavelength) 0.5–1.0. Streamlines and isothermal lines are presented to corresponding flow and thermal fields. Local and average Nusselt number distributions are presented. The obtained results are in good agreement with available numerical and experimental data.     

Laminar natural convection heat transfer from vertical plate to power-law fluid

Summary Approximate solutions for laminar natural convection heat transfer between a vertical plate and a power-law fluid with high Prandtl number were obtained using an integral method for cases with various types of boundary conditions. The results were found in good agreement with available experimental evidence.Nomenclature a exponent defined by equations (28) and (29) - A, B, C, D, E constants defined by equations (15) to (19) - C ₁, C ₂, M ₁, M ₂ coefficients for Nusselt number expression defined by (32b), (33b) - f temperature difference, equal to T _s–T - f ⁺ dimensionless temperature difference - g gravitational acceleration - Gr Grashof number defined by (25), (50) and (66), respectively - H heat flux at plate surface - h _x local heat transfer coefficient - K consistency index for Power-law fluid - k thermal conductivity of fluid - K ₁, K ₂ constants defined by (50) and (51) - L height of plate - n flow behavior index for Power-law fluid - P a quantity defined by (54a) - T temperature - T _s plate temperature - T temperature of the bulk of fluid - s constant given by (35) - u velocity component along x-direction - u _x maximum velocity induced by natural convection current, (10) - v velocity component along y-direction - x distance measured along direction parallel to that of gravitational force - x ⁺ dimensionless quantity, defined as x/L - y distance measured away from plate - Nu _x local Nusselt number - Nu _av average Nusselt number - Pr Prandtl number defined by (24) - T temperature difference according to boundary conditions - thermal diffusivity of fluid - coefficient of thermal expression of fluid - boundary layer thickness - ⁺ dimensionless boundary layer thickness - dimensionless velocity profile - dimensionless variable, defined as y/ - dimensionless temperature difference     

On the stability of the natural convection flow in a square cavity heated from the side

The stability of the steady laminar natural-convection flow of air (Prandtl number 0.71) and water (Prandtl number 7.0) in a square cavity is calculated by numerically solving the unsteady, two-dimensional Navier-Stokes equations. The cavity has a hot and cold vertical wall and either conducting or adiabatic horizontal walls. The flow looses its stability at a lower Rayleigh number in the case of conducting horizontal walls than in the case of adiabatic horizontal walls. The flow of water is more stable than the flow of air. Directly above the critical Rayleigh number the unsteady flow shows a single-frequency oscillation. Air in the case of adiabatic horizontal walls is an exception and shows two frequencies. The instabilities in the cavity seem to be related to well-known elementary instability mechanisms. In the case of conducting and adiabatic horizontal walls the instability seems to be related to a Rayleigh/Bénard and a Tollmien-Schlichting instability respectively. The second instability for air in the case of adiabatic horizontal walls seems to be related to an instability after a hydraulic jump.     

Laminar natural convection in a fully partitioned enclosure containing fluid with nonlinear thermophysical properties

The effects of a heat conducting partition on the laminar natural convection heat transfer and fluid flow were obtained by comparing the numerical and experimental results for a cubic enclosure without and with a partition. The two opposite vertical walls of the enclosure were isothermal at different temperatures. The working fluid was glycerol. The complete vertical partition, made of Plexiglass, was positioned in the middle of the enclosure. The visualizations of the velocity and temperature fields were obtained by using respectively, Plexiglass and liquid crystal particles as tracers. A middle plane perpendicular to the partition was numerically modeled. The steady two-dimensional model accounted for the variable thermophysical properties of the fluid. The finite volume method based on the finite difference approach was applied. The convective terms were approximated using a deferred correction central difference scheme. The velocity and temperature fields and the distribution of the local and average Nusselt numbers were found as a function of the Rayleigh (38 000 <Ra <369 000) and Prandtl (2700 < Pr < 7000) numbers.     

Transient laminar natural convection in a partitioned enclosure heated by uniform flux

Transient laminar natural convection in an enclosure partitioned by an adiabatic baffle is investigated numerically. The enclosure is heated by uniform heat flux from left wall and cooled from right wall which is kept at isothermal. A penalty finite element method with Newton-Raphson iteration algorithm and a backward difference scheme dealing with time term are adopted to solve governing equations. The effects of the baffle and Rayleigh number are found to be substantial on heat transfer mechanism during transient process. However, the variations of heat transfer mechanism occur mainly in the first one-third period of the time of transient, in spite of the present or not, or location of a baffle at the conditions of Rayleigh number being 10⁴ and 10⁶ in this study.Transiente, laminare freie Konvektion in einem von einer adiabaten Wand unterteilten Hohlraum wurde numerisch untersucht. Der Hohlraum wird von einem gleichmäßigen Wärmestrom von der linken Wand beheizt und von der rechten isothermen Wand gekühlt. Ein Finite-Element-Verfahren mit dem Newton-Raphson Iterationsalgorithmus und dem Differenzenverfahren mit Zeitterm sind übernommen worden, um bestehende Gleichungen zu lösen. Die Trennwand und die Rayleigh-Zahl sind für den Wärmeübergangsmechanismus während der transienten Phase von wesentlicher Bedeutung. Für den in dieser Studie untersuchten Bereich der Rayleigh-Zahl von 10⁴ und 10⁶ treten Veränderungen im Wärmeübergangsmechanismus hauptsächlich im ersten Drittel der transienten Phase auf und sind unabhängig von der Anwesenheit und Plazierung der Trennwand.     

Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects

The effect of double-diffusive natural convection of water in a partially heated enclosure with Soret and Dufour coefficients around the density maximum is studied numerically. The right vertical wall has constant temperature θ_c, while left vertical wall is partially heated θ_h, with θ_h > θ_c. The concentration in right wall is maintained higher than left wall (C_c < C_h) for case I, and concentration is lower in right wall than left wall (C_h > C_c) for case II. The remaining left vertical wall and the two horizontal walls are considered adiabatic. Water is considered as the working fluid. The governing equations are solved by control volume method using SIMPLE algorithm with QUICK scheme. The effect of the various parameters (thermal Rayleigh number, center of the heating location, density inversion parameter, Buoyancy ratio number, Schmidt number, and Soret and Dufour coefficients) on the flow pattern and heat and mass transfer has been depicted. Comprehensive Nusselt and Sherwood numbers data are presented as functions of the governing parameters mentioned above.     

A numerical simulation of combined radiation and natural convection heat transfer in a square enclosure heated by a centric circular cylinder

A numerical simulation of combined natural convection and radiation in a square enclosure heated by a centric circular cylinder and filled with absorbing-emitting medium is presented. The ideal gas law and the discrete ordinates method are used to model the density changes due to temperature differences and the radiation heat transfer correspondingly. The influence of Rayleigh number, optical thickness and temperature difference on flow and temperature fields along with the natural convection, radiation and total Nusselt number at the source surfaces is studied. The results reveal that the radiation heat transfer as well as the optical thickness of the fluid has a distinct effect on the fluid flow phenomena, especially at high Rayleigh number. The heat transfer and so the Nusselt number decreases with increase in optical thickness, while increases greatly with increase in temperature difference. The variation in radiation heat transfer with optical thickness and temperature difference is much more obvious as comparison with convection heat transfer.     

Turbulent natural convection in a square enclosure bounded by a solid wall with localized heating

Turbulent natural convection and conduction in a square enclosure bounded by a massive wall with a localized heating is numerically studied. The bounding solid wall has a relative thermal conductivity of 10 and a relative thickness of 0.1. Losses to the surroundings are specified using a Biot number of 500. Two-dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation and using the κ-ε model for turbulence are solved using finite difference method. Grids are generated in a nonuniform manner so that steep gradients near the wall regions are accounted for as required. Numerical solution is obtained for Ra numbers ranging from 10⁶ to 10¹³. The position of the source is also investigated. It is found that the heat transfer by convection is the highest when the heat source is located at the upper part of the cavity. The turbulent properties show also the same conclusion. Received on 4 November 1998