Convective heat transfer and temperature stratification in a sphere completely filled with a liquid,with a given heat flux

A number of articles have been devoted to the theoretical and experimental investigation of natural convection in spherical vessels completely filled with a liquid [1–6]. Analytical solutions are known, obtained by the expansion of the sought function in series in powers of the Rayleigh number (see, for example, [1]), valid for very small values of this number. A numerical solution of the nonlinear Boussinesq equations can be used to obtain solutions with larger Rayleigh numbers, but the existing data for spherical regions [2, 3] embrace a relatively narrow range of Rayleigh numbers. The experimental data with a given heat flux, published in [4–6], were obtained with relatively large Rayleigh numbers (Ra*=10⁹?10¹¹) and Prandtl numbers (P= 3?1500). Data on the characteristics of convection in spherical vessels are still not very numerous and, in a number of cases, contradictory. This relates, in particular, to the boundaries of unsteady-state conditions. The present article, continuing [7–9], expounds a method and gives the results of a calculation of convection in a sphere with a thinwalled shell, in a range of Rayleigh and Fourier numbers embracing the principal conditions of unsteady-state laminar convection with a given heat flux.     

Natural convection with surface radiation from a planar heat generating element mounted freely in a vertical channel

Experiments and numerical simulations have been conducted to study the conjugate heat transfer by natural convection and surface radiation from a planar heat generating element placed centrally between two adiabatic vertical plates. The relevant problem dependent parameters considered in this study are modified Rayleigh number, channel aspect ratio, stream-wise location of the heat generating element, and surface emissivities of the heat generating element and the adiabatic side plates. Experiments are conducted for different values of modified Rayleigh number ranging from 3.2 × 10⁵ to 1.6 × 10⁷ and surface emissivities 0.05, 0.55, 0.75 and 0.85. The interdependence between the heat transfer mechanism and the flow field under the influence of surface radiation on natural convection is explored and discussed. Experimental correlations for total and convective Nusselt number, and dimensionless temperature in terms of relevant parameters have been developed. The mathematical model governing the problem has been numerically solved using a commercial computational fluid dynamics package FLUENT 6.3 and the numerical predictions substantiate the experimental observations.     

Friction and heat transfer with simultaneous convection on a permeable surface

The velocity and heat transfer fields near a vertical permeable surface with simultaneous convection are investigated. A solution is found for the boundary layer equations with known laws of surface temperature and flow velocity change. The transformed boundary layer equations contain the parameter G/R², which determines the effect of free convection on friction and heat transfer for constrained motion. Calculations of friction and heat transfer as functions of draft (suction) with simultaneous convection are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 96–100, March–April, 1973.     

Turbulent penetrative convection with an internal heat source

An infinite fluid with a vertical cubic temperature profile in the absence of fluid motion is considered as a model for penetrative convection in which a central unstably stratified fluid layer is bounded above and below by stably stratified layers. Turbulence statistics from direct and large eddy numerical simulations for the mean temperature gradient, the velocity and temperature variances and the heat flux are presented for Rayleigh numbers R up to four orders of magnitude above critical. By means of a simplified second-moment closure, analytical scaling laws for the statistics are determined. For high Rayleigh numbers, the mean temperature gradient approaches zero in a central well-mixed layer, a reduced positive (stable) value in upper and lower partially mixed layers, and an unmixed value far above and below. The temperature variance is a factor of R^1/3 larger in the partially mixed layers compared to the well-mixed layer; the velocity variance and heat flux scales the same in both layers. Approximation of the three layers by a two layer model yields an estimate for the height of the mixed layer: the height decreases slowly with increasing Rayleigh number and at the highest Rayleigh number simulated is approximately 30% longer than the unstable layer in the absence of fluid motion.     

Natural convection flow of water near its density maximum in a rectangular enclosure having isothermal walls with heat generation

We consider unsteady laminar natural convection flow of water subject to density inversion in a rectangular cavity formed by isothermal vertical walls with internal heat generation. The top and bottom horizontal walls are considered to be adiabatic, whereas the temperature of the left vertical wall is assumed to be greater than that of the right vertical wall. The equations are non-dimensionalized and are solved numerically by an upwind finite difference method together with a successive over-relaxation (SOR) technique. The effects of both heat generation and variations in the aspect ratio on the streamlines, isotherms and the rate of heat transfer from the walls of the enclosure are presented. Investigations are performed for water taking Prandtl number to be Pr=11.58 and the Rayleigh number to be Ra=10⁵.     

Numerical analysis of natural convection in a thin-walled cylindrical vessel with hemispherical end-plates completely filled with liquid

The results are given of calculations of convection in a thin-walled cylindrical vessel with hemispherical end-plates over the range of Rayleigh and Fourier numbers that correspond to the main regimes of unsteady laminar convection in the presence of a given heat flux.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 204–203, September–October, 1984.     

Entropy generation in a square enclosure with partial heating from a vertical lateral wall

The entropy generation during the transient laminar natural convection in a square enclosure that is partially heated from a vertical lateral wall is numerically investigated. The active sites referring to the main irreversibility locations are determined. The Boussinesq approximation is used in the natural convection modelling. The effects of Prandtl (Pr) and Rayleigh (Ra) number combinations on the entropy generation are investigated. The study is restricted to the fluids of Prandtl number from 0.01 to 1.0, and Rayleigh numbers in the range of 10²–10⁸. It is found that the upper corner of the heated part of the side wall is the active site where the entropy generation initiates due to irreversibilities representing the energy loss.     

A numerical study of three-dimensional laminar natural convection in a vented enclosure

A three-dimensional numerical investigation of steady laminar natural convection in vented enclosures is carried out. A discrete flush-type heat source mounted on the substrate is used to simulate an electronic component. Four different vent locations are investigated. Combined natural convection in the air and conduction in the heat source, the substrate, and the enclosure walls are solved. Solutions are obtained for Rayleigh numbers ranging from 10⁴ to 10⁶, different substrate thermal conductivity ratios, and varied vent sizes. The calculation domain is extended beyond the cubic enclosure in x-, y-, and z-directions. Appropriate boundary conditions are prescribed on the extended computational domain. The resulting flow and temperature patterns are discussed. Also, the local and overall heat transfer from the heat source and the substrate, in terms of Nusselt numbers and the surface temperatures, are presented to illustrate the vent effects.     

Mean square velocities of bénard convection and heat transfer

The approximate formula K a^–2R(N–1), where a is a constant near 9 and R and N are the Rayleigh and Nusselt numbers, was proposed in [1] for the dimensionless kinetic energy K of convection in a horizontal layer of liquid. It is shown in the present paper that this expression is exact in linear and weakly nonlinear convection theory when the velocity and temperature fields are represented analytically [2–4]. The valuea is found to be 8.76 when the upper and lower boundaries of the layer are solid walls. The results are given of numerical calculations of the kinetic energy of the convection and the heat transfer in a wide range of Rayleigh numbers (up to 44 000) and Prandtl numbers (0.025 P 15). Analysis of the results shows that a is in fact a weak function of both R and P. If this is also the case at large R, it indicates a certain breaking of scaling of the mean convection characteristics at sufficiently large values of the Rayleigh number. It also indicates why laboratory experiments give values of n in the dependence N Rⁿ which are generally slightly less than the theoretical value n = 1/3.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 11–18, September–October, 1981.We should like to thank N. F. Vel'tishchev for providing first data of the numerical experiments of [13–15].     

Parametric excitation of Rayleigh convection in the presence of a variable heat flux on the free surface

The onset of Rayleigh convection in a semi-infinite fluid layer is investigated for a heat flux harmonically modulated along the normal to the surface of the fluid. The problem of the evolution of the velocity and temperature perturbations is solved numerically by means of a finite-difference method. The stability limits and the characteristics of the critical perturbations are determined as functions of the Prandtl numbers. The behavior of the critical Rayleigh number is studied for finite layer depths.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 40–44, November–December, 1996.     

Interferometric study of natural convection in rectangular air cavities of various orientations

The results of an experimental investigation into the temperature profiles and heat transfer associated with natural convection in rectangular air cavities are presented, the angle of inclination varying from 0 (heating at the bottom) to 180° (heating at the top). The range of Rayleigh numbers was R=2.68·10³–2.57·10⁵, and n=H/d=5.06–18.3. The investigation was carried out by an optical method, using an IZK-454 interferometer. For a horizontal orientation of the cavity the heat-transfer data satisfy the relation N=0.216 R^0.25, for a vertical orientation N=0.144 R^0.3h^–0.129, where N is the Nusselt number. In the region of an inclination of 30° the heat transfer passes through a maximum under all conditions studied.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 89–93, July–August, 1972.     

Effect of suction upon unsteady laminar free convection flow on a vertical infinite flat plate

Summary Unsteady laminar free convection flow past a vertical infinite flat plate subjected to suction is considered. Exact solutions of momentum and energy equations are obtained in two cases: (1) When the plate temperature is proportional to some power of time and (2) when the heat flux at the plate is proportional to some power of time. It is assumed that the suction velocity varies as (time)^–1/2. Expressions for the temperature and velocity profiles are obtained in closed forms in both the cases. Effect of suction on velocity, temperature, skin friction and rate of head transfer is studied for Prandtl numbers 0.02, 0.1, 0.72, 1 and 10.     

Transverse-roll global modes in a Rayleigh–Bénard–Poiseuille system with streamwise variable heating

The effect of a spatially inhomogeneous heating of the bottom wall in Rayleigh–Bénard–Poiseuille convection is studied for slow streamwise variations of the temperature profile. The problem is defined by the constant Reynolds number of the Poiseuille through flow, assumed to be low (typically 10), the constant Prandtl number, and the spatial evolution of the Rayleigh number , assumed to be subcritical everywhere except in a limited region around its single maximum . In this initial study, all spanwise inhomogeneities such as side walls or spanwise variable heating are neglected to obtain two-dimensional (transverse roll) global mode solutions by means of WKBJ asymptotics. The resulting frequency selection yields, at leading order, a global mode frequency equal to the local absolute frequency ω^t at the streamwise location where the Rayleigh number is maximum, with higher-order corrections for non-parallelism. These allow the determination of critical values of for global instability as a function of the profile of the local Rayleigh number and of Prandtl and Reynolds numbers.     

MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure

This study looks at MHD natural convection flow and heat transfer in a laterally heated enclosure with an off-centred partition. Governing equations in the form of vorticity–stream function formulation are solved using the polynomial differential quadrature (PDQ) method. Numerical results are obtained for various values of the partition location, Rayleigh, Prandtl and Hartmann numbers. The results indicate that magnetic field significantly suppresses flow, and thus heat transfer, especially for high Rayleigh number values. The results also show that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. The average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic field is applied. The results also show that flow and heat transfer have little dependence on the Prandtl number.     

Effect of adiabatic wall on the natural convection heat transfer from a wavy surface created by attached horizontal cylinders

The effect of a vertical adiabatic wall on the natural convection heat transfer from vertical array of attached cylinders, which can be considered as wavy surface, was investigated experimentally and numerically. The experiments were carried out using Mach-Zehnder interferometer and the commercial FLUENT code was used for numerical study. This paper focuses on the effect of wall-wavy surface spacing and Rayleigh number variation on the local and average free convection heat transfer coefficients from the each cylinder and the wavy surface. Rayleigh number ranges from 2400 to 10,000 and from 300,000 to 1,250,000 based on cylinder diameter and wavy surface height respectively. The local and average Nusselt numbers were determined for the different Rayleigh numbers, and the ratio of wall- wavy surface spacing to cylinder diameter 0.75, 1, 1.5, 2, 3, 4, 5, and ∞. Results are indicated with a single correlation which gives the average Nusselt number as a function of the ratio of the wall-wavy surface spacing to cylinder diameter and the Rayleigh numbers. There is an optimum distance between the wall and wavy surface in which the Nusselt number attain its maximum value. This optimum distance depends on the Rayleigh number.     

Direct numerical simulation of turbulent flow and heat transfer in a square duct with natural convection

In this paper, a direct numerical simulation of a fully developed turbulent flow and heat transfer are studied in a square duct with an imposed temperature difference between the vertical walls and the perfectly insulated horizontal walls. The natural convection is considered on the cross section in the duct. The numerical scheme employs a time-splitting method to integrate the three dimensional incompressible Navier-Stokes equation. The unsteady flow field was simulated at a Reynolds number of 400 based on the Mean friction velocity and the hydraulic diameter (Re _m = 6200), while the Prandtl number (Pr) is assumed 0.71. Four different Grashof numbers (Gr = 10⁴, 10⁵, 10⁶ and 10⁷) are considered. The results show that the secondary flow and turbulent characteristics are not affected obviously at lower Grashof number (Gr ≤ 10⁵) cases, while for the higher Grashof number cases, natural convection has an important effect, but the mean flow and mean temperature at the cross section are also affected strongly by Reynolds stresses. Compared with the laminar heat transfer at the same Grashof number, the intensity of the combined heat transfer is somewhat decreased.     

Experimental investigation of natural convection in a vertical rib-roughened channel with asymmetric heating

An experimental study in an open-ended vertical channel is carried out in order to describe the fluid dynamics and heat transfer of transient free convection inside a vertical rib-roughened channel asymmetrically heated at various uniform heat fluxes (650, 700, and 780 W/m²) corresponding to various modified Rayleigh numbers (3.65 × 10⁶, 3.93 × 10⁶ and 4.4 × 10⁶). Two ribs are symmetrically located on each wall. The investigations focused more specifically on the influence of the ribs positions inside the channel and the modified Rayleigh number (Ra^*) both in steady-state regime and during the transitional phase occurring just after the start of the heating on the flow structure and the heat transfer performance. The results showed the appearance of large-scale flow instabilities which will develop and propagate until the development of the pocket-like vortex (reversed flow). Also, the formation and breakup of recirculation eddies, vortex banishment, besides that a separation and shifting of the boundary layer from one wall to another are identified. The best position of the ribs for heat extraction depends on the magnitude of the Rayleigh number. In that case, the top position is the optimal position for the small and the moderate modified Rayleigh numbers.