Showing papers on "Rayleigh number published in 1972"
TL;DR: In this article, the complex nature of the natural convection phenomena in enclosures is discussed and the boundary value problem is formulated, assuming that the motion is two-dimensional and steady, the fluid is incompressible and frictional heating is negligible.
Abstract: Publisher Summary This chapter discusses the complex nature of the natural convection phenomena in enclosures. It discusses the two basic configurations of natural convection— that is, a rectangular cavity and a horizontal circular cylinder. In rectangular cavities, consideration is given to the two-dimensional convective motion generated by the buoyancy force on the fluid in a rectangle and to the associated heat transfer. The two long sides are vertical boundaries held at different temperatures and the short sides can either be heat conducting or insulated. Particular attention is given to the different flow regimes that can occur and the heat transfer across the fluid space between the two plane parallel vertical boundaries. Although heat transfer by radiation may not be negligible it is independent of the other types of heat transfer and can be fairly accurately calculated separately. To formulate the boundary value problem that describes this phenomena it is assumed that: (a) the motion is two-dimensional and steady, (b) the fluid is incompressible and frictional heating is negligible, and (c) the difference between the hot wall and cold wall temperatures is small relative to the absolute temperatures of the cold wall. In horizontal circular cylinder, consideration is given to the large Rayleigh number flow with the Prandtl number large and the Grashof number of unit order of the magnitude.
382 citations
TL;DR: In this paper, a numerical and experimental investigation of the developing laminar free convection heat transfer in vertical parallel plate channels with asymmetric heating is presented, where the Nusselt number characterizing the total heat transfer to the fluid is found to be related to the Rayleigh number very nearly by a universal curve for all ratios of wall temperature differences.
330 citations
TL;DR: In this article, the true critical Rayleigh number for the onset of convective flow of a fluid in a rectangular box of porous material heated from below is found for various box geometries.
Abstract: The true critical Rayleigh number for the onset of a convective flow of a fluid in a rectangular box of porous material heated from below is found for various box geometries. The preferred cellular mode of the motion at Rayleigh numbers just above the critical is determined. In contrast with the established results for the similar problem in a continuous fluid, the roll (a cell with only two nonzero velocity components) is not the only cellular mode and the roll axis direction is such that there is the greatest degree of “squareness” in the cross section of each roll. The invalidity of a frequently used form of Darcy's law and the present form of the energy method for the stability of flows in which fluid crosses the boundaries is discussed.
250 citations
TL;DR: In this article, the instability of convection rolls in a fluid layer heated from below is investigated for stress-free boundaries in the limit of small Prandtl number, and it is shown that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value.
Abstract: The instability of convection rolls in a fluid layer heated from below is investigated for stress-free boundaries in the limit of small Prandtl number. It is shown that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value. The instability corresponds to the generation of vertical vorticity, a mechanism which is likely to operate in the case of a variety of roll-like motions. In all aspects in which the theory can be related to experiments, reasonable agreement with the observations is found.
211 citations
TL;DR: In this article, the authors measured energy transport at Rayleigh numbers up to 675 times the critical (linear stability theory) value in a layer of dilute electrolyte bounded horizontally by two rigid planes of constant and equal temperature; Joule heating by an alternating current passing horizontally through the layer provides the volumetric energy source.
Abstract: Energy transport at Rayleigh numbers up to 675 times the critical (linear stability theory) value is measured in a layer of dilute electrolyte bounded horizontally by two rigid planes of constant and equal temperature; Joule heating by an alternating current passing horizontally through the layer provides the volumetric energy source. Horizontally averaged temperature profiles are determined optically. Mean temperature distributions are asymmetric at elevated Rayleigh numbers, the energy transport at the upper boundary being more than twice that at the lower boundary. Three regimes of flow are identified and discrete transitions in the energy transport appear to exist when the flow is turbulent. Extrapolation of the data to the conduction value of the Nusselt number yields a critical Rayleigh number which is within + 10·7% of linear theory values. No subcritical convection is observed when finite amplitude disturbances are introduced into the fluid at a Rayleigh number between the critical values predicted by the linear stability theory and energy theory respectively.
176 citations
TL;DR: In this article, the dependence of the Nusselt number on the Rayleigh number was examined to the sixth order using an expansion for the RPN proposed by Kuo (1961).
Abstract: For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcy's law.
146 citations
TL;DR: In this article, the linear stability of a rotating, electrically conducting viscous layer, heated from below and cooled from above, and lying in a uniform magnetic field is examined, using the Boussinesq approximation.
Abstract: The linear stability of a rotating, electrically conducting viscous layer, heated from below and cooled from above, and lying in a uniform magnetic field is examined, using the Boussinesq approximation. Several orientations of the magnetic field and rotation axes are considered under a variety of different surface conditions. The analysis is, however, limited to large Taylor numbers, T , and large Hartmann numbers, M . (These are non-dimensional measures of the rotation rate and magnetic field strength, respectively.) Except when field and rotation are both vertical, the most unstable mode at marginal stability has the form of a horizontal roll whose orientation depends in a complex way on the directions and strengths of the field and angular velocity. For example, when the field is horizontal and the rotation is vertical, the roll is directed parallel to the field, provided that the field is sufficiently weak. In this case, the Rayleigh number, R (the non-dimensional measure of the applied temperature contrast) must reach a critical value, R c , which is O ( T 2/5 ) before convection will occur. If, however, the field is sufficiently strong [ T = O ( M 4 )], the roll makes an acute angle with the direction of the field, and R c = O ( T 1/2 ), i.e. the critical Rayleigh number is much smaller than when the magnetic field is absent. Also, in this case the mean applied temperature gradient and the wavelength of the tesselated convection pattern are both independent of viscosity when the layer is marginally stable. Furthermore, the Taylor-Proudman theorem and its extension to the hydromagnetic case are no longer applicable even qualitatively. Over the interior of the layer, however, the Coriolis forces to which the convective motions are subjected are, to leading order, balanced by the Lorentz forces. The results obtained in this paper have a bearing on the possibility of a thermally driven steady hydromagnetic dynamo.
139 citations
TL;DR: In this article, an algorithm was developed for the finite-difference computation of hydrodynamic stability and natural convection in non-Newtonian fluids heated from below, and the results were found to be independent of the assumed initial state.
Abstract: An algorithm was developed for the finite-difference computation of hydrodynamic stability and natural convection in non-Newtonian fluids heated from below. Test calculations were carried out for fluids whose viscosity characteristics are described by the Ostwald-de Waele (power-law) and Ellis models and for roll-cells with both rigid and dragless vertical boundaries. The effects of time-step and grid-size were tested thoroughly. The results were found to be independent of the assumed initial state. The computed values of the Nusselt number and the critical Rayleigh number for Newtonian fluids agree well with prior experimental results. The computations for the Ostwald-de Waele model indicate that the approximate solution of Tien, Tsuei, and Sun may underestimate the critical Rayleigh Number.
131 citations
TL;DR: In this article, a finite difference solution of the equations describing transient natural convection in porous media is presented, and the linearized equations are solved to provide an estimate of the number of possible convective modes as a function of the Rayleigh number.
112 citations
TL;DR: In this article, an extension of the analysis of Nield is made to more completely characterize the onset of convection in an infinite horizontal porous medium stratified by temperature and concentration.
Abstract: An extension of the analysis of Nield is made to more completely characterize the onset of convection in an infinite horizontal porous medium stratified by temperature and concentration. Comparisons are made with thermohaline convection in Newtonian fluids. Major differences lie in the concentration Rayleigh number dependence of the wavenumber at the marginal state of overstability and the dependence of the horizontal wavenumber in the “salt finger” region of stationary convection on both temperature and concentration Rayleigh numbers. Suggestions of geological applications and laboratory verification using a Hele‐Shaw cell are presented.
110 citations
TL;DR: The average roll diameter in Rayleigh convection for 2000 < R < 31000, where R is the Rayleigh number, has been measured from photographs of three convecting fluids: air, water and a silicone oil with a Prandtl number σ of 450.
Abstract: The average roll diameter in Rayleigh convection for 2000 < R < 31000, where R is the Rayleigh number, has been measured from photographs of three convecting fluids: air, water and a silicone oil with a Prandtl number σ of 450. For air the average dimensionless roll diameter was found to depend uniquely upon R and to increase especially rapidly in the range 2000 < R < 8000. The fluids of larger σ exhibited strong hysteresis but also had average roll diameters tending to increase with R. The increase in average roll diameter with R tended to decrease with σ. Through use of two-dimensional numerical integrations for the case of air it was found that the increase in average roll diameter with R provides an explanation for the usual discrepancy in heat flux observed between experiment and two-dimensional numerical calculations which prescribe a fixed wavelength.
TL;DR: In this article, the heat transfer properties of a vertical array of heated cylinders are obtained in steady state natural convection, showing that the surface temperature increases with elevation in the array for closely spaced arrays, but decreases with elevation when the spacing is sufficiently large.
TL;DR: In this paper, the initiation of natural convection in a fluid confined above and below by rigid perfectly conducting surfaces and laterally by rigid, perfectly insulating vertical walls that form a rectangular shape is examined.
TL;DR: In this article, the authors deal with natural convection heat transfer from a non-isothermal vertical flat plate immersed in a temperature stratified environment and show that approximations based on the local temperature difference can introduce large errors into the prediction of surface heat transfer rates.
TL;DR: In this article, the formation of convective cells in a fluid between two horizontal rigid boundaries with time-periodic temperature distribution is studied by the use of the Floquet theory, and numerical results for the critical Rayleigh number are given for a Prandtl number of 0·73 (air) and for various values of the frequency and magnitude of the primary temperature oscillation.
Abstract: The formation of convective cells in a fluid between two horizontal rigid boundaries with time-periodic temperature distribution is studied by the use of the Floquet theory. Numerical results for the critical Rayleigh number are given for a Prandtl number of 0·73 (air) and for various values of the frequency and magnitude of the primary temperature oscillation. Some numerical results for a Prandtl number of 7·0 (water) are also given. The most striking feature of the results is that the disturbances (or convection cells) oscillate either synchronously or with half frequency.
TL;DR: In this article, the effects of single scattering albedo, optical thickness, and conduction-to-radiation parameter on temperature distribution in the boundary layer and heat transfer at the wall were investigated.
TL;DR: In this article, the authors performed finite difference computations for convection in a square cavity at Rayleigh number ∼O(O(106) for a variety of dynamical boundary conditions, Rayleigh numbers, and Prandtl numbers.
Abstract: Seven finite difference computations for convection in a square cavity at Rayleigh number ∼O(106) have been carried out for a variety of dynamical boundary conditions, Rayleigh numbers, and Prandtl numbers. The results indicate that various horizontal dynamical boundary conditions virtually make no difference to the main boundary‐layer flow although free vertical boundaries double the velocity in the boundary layer adjacent to them. The scaled fields in the vertical boundary layers are independent of the Rayleigh number, and to a large extent, independent of the Prandtl number, when the two vertical boundary layers are distinct. The results compare reasonably well with Elder's experiments.
TL;DR: In this paper, an experimental investigation of free convection heat transfer from heated spheres to water is reported, and the experimental data extend over a wide range of Rayleigh number, thus covering the laminar, transition, and beginning of the turbulent regimes.
TL;DR: In this article, a quasi-two-dimensional, free convection heat transfer from a heated horizontal plate of Rayleigh number of the order of 107 in air has been made, and the velocity and temperature fields near the downward-facing surface have been measured.
TL;DR: In this article, the formation and growth of horizontal layered convection cells in a density stratified solution of salt water subject to an impulsively applied lateral temperature gradient is investigated with physical and numerical experiments.
Abstract: The formation and growth of horizontal layered convection cells in a density stratified solution of salt water subject to an impulsively applied lateral temperature gradient is investigated with physical and numerical experiments. Results indicate that lyers are induced by two mechanisms. One is the successive formation of layers due to the presence of the top and bottom boundaries. The other is the spontaneous occurrence of layers when a suitably defined Rayleigh number exceeds a critical value. It is found that well established layers are homogeneous in temperature and salinity and are separated by sharp gradients in density. Lateral heat transfer is of a periodic nature. Numerical experiments were carried out for finite and infinite geometry cases. For the finite geometry case, convection cells are generated successively inward from the horizontal boundaries. For the infinite geometry case, periodic conditions in the vertical direction are assumed. With continuous input of small perturbations,...
TL;DR: In this paper, a general formulation valid for all Prandtl numbers is presented and the limiting case of large PrandTL number is approached by a numerical method, where the typical developments of temperature profile, wall temperature and secondary flow in the thermal entrance region are presented for the case of square channel γ = 1.
TL;DR: In this paper, the stability of a two-component fluid layer heated from below is examined taking into account the concentration gradient due to thermal diffusion, and an approximate solution is proposed for dilute solutions.
Abstract: In connection with recent experimental results, the stability of a two‐component fluid layer heated from below is examined taking into account the concentration gradient due to thermal diffusion. With the use of a variational method (local potential technique) developed by Glansdorff and Prigogine, an approximate solution is proposed for dilute solutions. The critical Rayleigh number increases for negative thermal diffusion factors and decreases for positive ones.
TL;DR: In this paper, a model of thermal convection of a Boussinesq fluid in an equatorial annulus of a rotating spherical shell is presented, where the convection induces and maintains differential rotation and meridian circulation.
Abstract: We present extensive numerical calculations for a model of thermal convection of a Boussinesq fluid in an equatorial annulus of a rotating spherical shell. The convection induces and maintains differential rotation and meridian circulation. The model is solved for an effective Prandtl number P = 1, with effective Taylor number T in the range 102
TL;DR: In this article, the authors measured the time-dependent mass transfer rates in the transition and turbulent regimes, and the measured local laminar mass transfer coefficients agreed with analytical predictions, both for vertical and inclined surfaces.
TL;DR: In this article, the authors examined the two-dimensional convective motion of a nonrotating incompressible Boussinesq fluid heated non-uniformly from below.
Abstract: This paper examines the two-dimensional convective motion of a nonrotating incompressible Boussinesq fluid heated non-uniformly from below. The fluid container is rectangular; the side and top boundaries are insulating and rigid. A linear temperature field is maintained along the bottom boundary. Using the DuFort-Frankel scheme for diffusion and the Arakawa scheme for advection, the governing vorticity and temperature equations are integrated numerically for two cases, the first having a stress-free bottom boundary and the second having a constant stress along the bottom boundary. In the first case, a single convective cell develops; an intense buoyant jet of fluid rises from the warmer section of the bottom while there is a more uniform sinking motion over the cooler section of the bottom. The cell asymmetry, the circulation, and the convective heat transfer increase markedly with increasing Rayleigh number (based here on fluid properties, cell height, and the horizontal temperature difference a...
TL;DR: In this paper, the authors focused on the "delay time" between the application of current to the wire and the beginning of observable convection, i.e., the time when a thin horizontal wire is heated by passing an electrical current through it.
TL;DR: In this article, a model of thermal convection under a central force field has been constructed using a strong alternating electric field gradient in a dielectric liquid, which is governed by an electrical Rayleigh number.
Abstract: A laboratory model of thermal convection under a central force field has been constructed using a strong, alternating electric field gradient in a dielectric liquid Both the electric field gradient and a temperature gradient are maintained between concentric vertical cylinders The onset of thermal convection is detected by heat transfer and temperature measurements It is governed by an electrical Rayleigh number, in which the electric force replaces gravity Marginal stability analysis gives a critical electrical Rayleigh number in agreement with the experimentally determined value
TL;DR: In this paper, the structure of finite-amplitude thermal convection within a self-gravitating fluid sphere with uniform heat release was determined for a fixed-surface boundary condition.
Abstract: Finite-difference calculations have been carried out to determine the structure of finite-amplitude thermal convection within a self-gravitating fluid sphere with uniform heat release. For a fixed-surface boundary condition, single-cell convection breaks up into double-cell convection at a Rayleigh number of 30,000, at a Rayleigh number of 500,000 four-cell convection is observed. With a free-surface boundary condition only single cell convection is obtained up to a Rayleigh number of 5,000,000.