Natural convection in an inclined enclosure containing internal energy sources and cooled from below

Natural convection in an inclined enclosure from below and containing internally heated fluid has been investigated using a finite difference calculation procedure. Results have been obtained for Rayleigh number values up to 10⁶ and for inclination angles of 30 and 60°. For internal Rayleigh numbers that are much larger than the external Rayleigh number, the flow rises in the interior and moves down both the hot and cold walls. On the other hand, if the external Rayleigh number has a larger magnitude, the flow moves upwards along the hot surface and downwards along the cold surface. For the latter situation, the inner core is multicellular in nature at large external Rayleigh numbers. The average heat flux ratio along the cold surface (convective heat flux/corresponding conduction heat flux) increases with increasing external Rayleigh number and decreasing internal ratio is non-monotonic in nature. The heat flux ratio along both surfaces is observed to be strongly dependent on the inclination angle at high external Rayleigh numbers. A maximum in the local heat flux along the cold surface is obtained in the vicinity of x/L = 1 where hot fluid, either from the interior or directly from the opposite hot wall, meets the surface. Along the hot wall, a maximum in the heat flux ra flo     

Natural convection in vertical cavities with internal heat generating porous medium

Steady natural convection heat transfer in a two dimensional cavity filled with a uniform heat generating, saturated porous medium has been studied. The boundary conditions were: Two isothermal walls at different temperatures, two horizontal adiabatic walls. The aspect ratio was varied from 0.1 to 10 and the Rayleigh number from 10⁰ to 10⁸. The results are presented in terms of the isotherms and stream functions, the temperature variation and maximum temperature in the cavity and heat transfer from the vertical walls. The study indicates that asymmetric vertical boundary conditions with _h >0 has an important effect on the temperature and flow fields as well as on the heat transfer characteristics of the cavity with highly asymmetric results. Various heat transfer modes are identified dependent on the Rayleigh number and the aspect ratio.Es wurde stetiger Wärmeübergang durch freie Konvektion in einen zweidimensionalen Hohlraum, gefüllt mit gleichmäßig wärmeerzeugendem porösen Medium, untersucht. Dabei wurden folgende Randbedingungen festgelegt: zwei isotherme Wände unterschiedlicher Temperatur, zwei horizontale adiabate Wände. Das Verhältnis vonH/L wurde zwischen 0,1 und 10, die Rayleigh-Zahl zwischen 10⁰ und 10⁸ variiert. Die Ergebnisse werden durch Terme der Temperaturverläufe, der maximalen Temperatur im Hohlraum, der Isothermen und der Strömungsfunktion, sowie der Wärmeübertragung der vertikalen Wände dargestellt. Die Studie zeigt, daß asymmetrische Randbedingungen mit _h >0 einen großen Einfluß auf Temperatur- und Strömungsfelder, als auch auf die Wärmeübertragungskennzahlen des Hohlraumes mit stark asymmetrischen Ergebnissen haben. Die verschiedenen Arten der Wärmeübertragung werden in Abhängigkeit von der Rayleigh-Zahl und dem Verhältnis vonH/L beschrieben.     

Natural convection in partially cooled tilted cavities

Two-dimensional numerical simulations of laminar natural convection in a partially cooled, differentially heated inclined cavities are performed. One of the cavity walls is entirely heated to a uniformly high temperature (heat source) while the opposite wall is partially cooled to a lower temperature (heat sink). The remaining walls are adiabatic. The tilt angle of the cavity is varied from 0° (heated from left) to −90° (heated from top). The fast false implicit transient scheme (FITS) algorithm, developed earlier by the same authors, is modified to solve the derived variables vorticity-streamfunction formulation. The effects of aspect ratio (AR), sink–source ratio and tilt angle on the average Nusselt number are examined through a parametric study; solutions are obtained for two Grashof numbers, 10⁵ and 10⁷. Flow patterns and isotherms are used to investigate the heat transfer and fluid flow mechanisms inside the cavity. © 1998 John Wiley & Sons, Ltd.     

Natural convection in inclined two dimensional rectangular cavities

Steady two-dimensional natural convection in fluid filled cavities is numerically investigated. The channel is heated from below and cooled from the top with insulated side walls and the inclination angle is varied. The field equations for a Newtonian Boussinesq fluid are solved numerically for three cavity height based Rayleigh numbers, Ra = 10⁴, 10⁵ and 10⁶, and several aspect ratios. The calculations are in excellent agreement with previously published benchmark results. The effect of the inclination of the cavity to the horizontal with the angle varying from 0° to 180° and the effect of the startup conditions on the flow pattern, temperature distribution and the heat transfer rates have been investigated. Flow admits different configurations at different angles as the angle of inclination is increased depending on the initial conditions. Regardless of the initial conditions Nusselt number Nu exhibits discontinuities triggered by gradual transition from multiple cell to a single cell configuration. The critical angle of inclination at which the discontinuity occurs is strongly influenced by the assumed startup field. The hysteresis effect previously reported is not always present when the calculations are reversed from 90° to 0°. A comprehensive study of the flow structure, the Nu variation with varying angle of inclination, the effect of the initial conditions and the hysteresis effect are presented.     

Non-linear convection in a porous medium with internal heat sources

The paper studies non-linear thermal convection in a horizontal porous layer of fluid with nearly insulating boundaries and in the presence of internal heat sources. Square and hexagonal cells are found to be the only possible stable convection cells. Finite amplitude instability could exist for some particular forms of an internal heat source Q. For a uniform Q, the preferred flow pattern is that of hexagons for amplitude ε smaller than some critical value ε_c, while both squares and hexagonal cells are stable for ε ? ε_c. The convective motion is downward at the hexagonal cell's centers. For a non-uniform Q, the qualitative features of thermal convection depend on the actual form Q. In particular, a non-uniform Q can increase or decrease the cell's size and the critical Rayleigh number at the onset of convection, and stabilize or destabilize the convective motion in the form of hexagonal cells with either upward or downward motion at the cell's centers.     

Natural convection and conduction heat transfer in open shallow cavities with bounding walls

A numerical study has been carried out on inclined open shallow cavities, which are formed by a wall and horizontal fins. Constant heat flux is applied on the surface of the wall inside the cavity while its other surface was kept isothermal. The wall and the fins are conductive. Conjugate heat transfer by natural convection and conduction is studied by numerically solving equations of mass, momentum and energy. Streamlines and isotherms are produced, heat and mass transfer is calculated. A parametric study is carried out using following parameters: Rayleigh number from 10⁶ to 10¹², conductivity ratio from 1 to 60, open cavity aspect ratio from 1 to 0.125, dimensionless end wall thickness from 0.05 to 0.20, horizontal walls from 0.01 to 0.15 and inclination of the end wall from 90° to 45°. It is found that the volume flow rate and Nusselt number are a decreasing function of the cavity aspect ratio, horizontal fin thickness and conductivity ratio. They are an increasing function of end wall thickness and inclination angle, except in the latter case optima exist at high Rayleigh numbers.     

Buoyancy-driven flow in an enclosure containing time periodic internal sources

A computational study is carried out to investigate the effect of the sinusoidally driven heat source on the fluid flow and heat transfer within a two-dimensional square cavity. The cavity, which has solid walls of constant temperature, is filled with a fluid including uniformly distributed internal heat source. In addition, the effects of the different periods of the sinusoidally driving heat source on heat transfer are investigated and presented as figures.     

Natural convection in a square enclosure with an internal isolated vertical plate

Numerical computations were performed for the average Nusselt number at an internal vertical plate situated in a square enclosure, with the inner plate and the bounding wall of the enclosure maintained at uniform but different temperatures. Natural convection occurred in the air which occupied the enclosure space. The position of the inner vertical plate within the enclosure was varied parametrically. The plate height-cavity height ratio was 0.513. For narrow distance between the inner plate and the bounding wall the inner plate Nusselt number was enhanced. Aside from this, the plate average Nusselt number was remarkably insensitive to the plate position. The effect of the Rayleigh number on the velocity and temperature fields and local Nusselt numbers are also discussed. The agreement between the predicted flow pattern forRa=1.1×10⁶ and the flow visualization result was reasonably good.

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Natürliche Konvektion in einem quadratischen Horizontalschacht, der eine freistehende, senkrechte Platte enthält

Zusammenfassung Eine numerische Untersuchung liefert mittlere Nußelt-Zahlen an einer, in einem quadratischen Horizontalschacht freistehenden, senkrechten Platte, wobei deren Temperatur und die der umgebenden Wände jeweils konstant gehalten werden. Im Luftraum dazwischen stellte sich freie Konvektion ein. Die Position der Platte war veränderlich, ihre Höhe blieb mit 51.3% der Schachthöhe konstant. Rückte die Platte nahe an eine Schachtwand, so erhöhte sich die Nußelt-Zahl auf der dieser zugewandten Seite, während die Gesamt-Nußelt-Zahl bezüglich der Platte fast konstant bleibt. Es wird auch der Einfluß der Rayleigh-Zahl auf das Geschwindigkeitsund Temperaturfeld diskutiert. BeiRa=1.1·10⁶ stimmten die Ergebnisse aus der Berechnung gut mit den experimentellen Befunden einer Strömungsvisualisation überein.

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Nomenclature a distance between vertical plate and side-wall of enclosure thermal diffusivity (in definition ofu _r) - b distance between vertical plate and bottom of enclosure - g gravitational acceleration - G characteristic flow rate - H height of vertical plate - k thermal conductivity - k _f fluid thermal conductivity - K relative thermal conductivity,k/k _f - L width of square enclosure - M _res mass residual - Nu local Nusselt number - Nu _m average Nusselt number - Nu _L local Nusselt number of left side of vertical plate - Nu _R local Nusselt number of right side of vertical plate - Nu _B local Nusselt number of bottom side of vertical plate - Nu _T local Nusselt number of top side of vertical plate - p effective pressure - P dimensionless pressure,P=p/[(Ra Pr)(a/H)²] - Pr Prandtl number - Ra Rayleigh number,Ra=gTH ³ Pr/ ² - T temperature - T _i temperature of internal plate - T _o temperature of enclosure surface - u, v velocity components inx-, y-direction - U, V dimensionless velocities,U=u/u _r, V=v/u_r - u _r reference velocity,u _r=(Ra Pr)^1/2/(a/H) - X, {iyY} dimensionless coordinates,X=x/H, Y=y/H Greek symbols heat transfer coefficient - volume expansion coefficient - thickness of plate - kinematic viscosity - density - dimensionless temperature, (T _i–T)/(T _i–T_o)     

Natural convection in a rotating anisotropic porous layer with internal heat generation

In this article, linear and nonlinear thermal instability in a rotating anisotropic porous layer with heat source has been investigated. The extended Darcy model, which includes the time derivative and Coriolis term has been employed in the momentum equation. The linear theory has been performed by using normal mode technique, while nonlinear analysis is based on minimal representation of the truncated Fourier series having only two terms. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. Effects of parameters on critical Rayleigh number has also been investigated. A weak nonlinear analysis based on the truncated representation of Fourier series method has been used to find the Nusselt number. The transient behavior of the Nusselt number has also been investigated by solving the finite amplitude equations using a numerical method. Steady and unsteady streamlines, and isotherms have been drawn to determine the nature of flow pattern. The results obtained during the analysis have been presented graphically.     

Three dimensional simulations of penetrative convection in a porous medium with internal heat sources

The problem of penetrative convection in a fluid saturated porous medium heated internally is analysed. The linear instability theory and nonlinear energy theory are derived and then tested using three dimensions simulation.Critical Rayleigh numbers are obtained numerically for the case of a uniform heat source in a layer with two fixed surfaces. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. Our results show that the linear threshold accurately predicts the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Mixed convection and conduction heat transfer in open cavities

A numerical study has been carried out on mixed convection and conduction in open cavities. The study covers the Rayleigh number from 10 to 10⁶, the Reynolds number from 1 to 10³ andk _r =k _w /k _a from 1 to 100 forPr=0.72 (air) and cavity aspect ratioB from 0.8 to 1.3. The flow is assumed to be laminar and two-dimensional. The density variation is taken into account by the Boussinesq approximation. The controlvolume approach is used for solving the governing equations of conjugate heat transfer involving conduction in the walls. Streamlines and isotherms in the system are produced and the results are represented in terms of the Nusselt number as function of other parameters. Correlations are derived to calculate heat transfer through the cavity openings.     

Core stratification effects on natural convection in inclined cavities

A numerical and experimental study has been made on the flow and heat transfer in inclined air-filled cavities with aspect ratios 1–18 at Ra numbers from 2·10⁴–5·10⁵ and angles of inclination from 40 to 90°. Core stratification influences the flow. Due to this there arises a torque with two components depending on angle of inclination. On basis of the two torques the computed effects on flow and temperature fields can be explained. For the heat transfer a scaling law could be derived. Experimental data validate the numerical studies.     

Natural convection of nano-fluids

Fluids with nano size solid particles suspended in them have been given the name nano-fluid which in recent studies have shown tremendous promise as heat transfer fluids. However, before suggesting such fluids for applications a thorough knowledge of physical mechanism of heat transfer in such fluids is wanted. The present study deals with one such aspect of natural convection of nano fluids inside horizontal cylinder heated from one end and cooled from the other. An apparently paradoxical behaviour of heat transfer deterioration was observed in the experimental study. Nature of this deterioration and its dependence on parameters such as particle concentration, material of the particles and geometry of the containing cavity have been investigated. The fluid shows characters distinct from that of common slurries.     

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.     

Natural convection in inclined partitioned enclosures

The problem of steady, laminar, natural convective flow of a viscous fluid in an inclined enclosure with partitions is considered. Transverse gradient of temperature is applied on the two opposing regular walls of the inclined enclosure while the other walls are maintained adiabatic. The problem is formulated in terms of the vorticity-stream function procedure. A numerical solution based on the finite volume method is obtained. Representative results illustrating the effects of the enclosure inclination angle and the degree of irregularity on the contour maps of the streamlines and temperature are reported and discussed. In addition, results for the average Nusselt number at the heated wall of the enclosure and the difference of extreme stream-function values are presented and discussed for various Rayleigh numbers, inclination angles and dimensionless partition heights.     

Thermocapillary instability of the equilibrium of a flat layer containing internal heat sources

In the absence of body forces, a factor which has a strong influence on the equilibrium stability of a nonuniformly heated liquid is the dependence of the coefficient of surface tension on the temperature and the thermocapillary effect generated by it. If the equilibrium temperature gradient is sufficiently great, then the presence of the thermocapillary forces on the free surface can lead to the occurrence of convective motion. The monotonie instability of the equilibrium of a flat layer was investigated in [1–3]. Analysis of nonmonotonic disturbances [4] showed that in the case of an undeformable free surface there is no oscillatory instability. In [5] it was found that oscillatory instability is possible if there is a nonlinear dependence of the coefficient of surface tension on the temperature. The present paper is devoted to numerical investigation of the equilibrium stability of a flat layer with respect to arbitrary disturbances. It is shown that for a deformable free boundary there appears an additional neutral curve, which corresponds to monotonie capillary instability. In addition, when the capillary convection mechanism is taken into account, there appears an oscillatory instability, which becomes the most dangerous in the region of small Prandtl and wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–31, March–April, 1991.I thank V. K. Andreev for a helpful discussion of the work.     

Natural convection in vertical permeable space

This article describes the natural convective heat transfer in a vertical space insulated with a permeable material. A numerical solution to the problem is given for Ra₀ ? 200, which is the range of interest to building technological applications. The variables and boundary conditions and their influence upon the convective heat transfer are discussed. Comparisons are made with previously published results.