Showing papers on "Rayleigh number published in 1997"
TL;DR: In this article, a two-component lattice Boltzmann equation (LBE) method was used to simulate Rayleigh-B\'enard convection in two and three dimensions.
Abstract: Rayleigh-B\'enard convection is numerically simulated in two and three dimensions using a recently developed two-component lattice Boltzmann equation (LBE) method. The density field of the second component, which evolves according to the advection-diffusion equation of a passive scalar, is used to simulate the temperature field. A body force proportional to the temperature is applied, and the system satisfies the Boussinesq equation except for a slight compressibility. A no-slip, isothermal boundary condition is imposed in the vertical direction, and periodic boundary conditions are used in horizontal directions. The critical Rayleigh number for the onset of the Rayleigh-B\'enard convection agrees with the theoretical prediction. As the Rayleigh number is increased higher, the steady two-dimensional convection rolls become unstable. The wavy instability and aperiodic motion observed, as well as the Nusselt number as a function of the Rayleigh number, are in good agreement with experimental observations and theoretical predictions. The LBE model is found to be efficient, accurate, and numerically stable for the simulation of fluid flows with heat and mass transfer.
519 citations
TL;DR: In this article, a generalised non-Darcian porous medium model for natural convective flow has been developed taking into account linear and non-linear matrix drag components as well as the inertial and viscous forces within the fluid.
498 citations
TL;DR: In this paper, an experimental study of Rayleigh-Benard convection in the strongly turbulent regime is presented, where the authors report results obtained at low Prandtl number (in mercury, Pr = 0.025) and compare them with results at Pr∼1.
Abstract: An experimental study of Rayleigh–Benard convection in the strongly turbulent regime is presented. We report results obtained at low Prandtl number (in mercury, Pr = 0.025), covering a range of Rayleigh numbers 5 × 106 < Ra < 5 × 109, and compare them with results at Pr∼1. The convective chamber consists of a cylindrical cell of aspect ratio 1.Heat flux measurements indicate a regime with Nusselt number increasing as Ra0.26, close to the 2/7 power observed at Pr∼1, but with a smaller prefactor, which contradicts recent theoretical predictions. A transition to a new turbulent regime is suggested for Ra ≃ 2 × 109, with significant increase of the Nusselt number. The formation of a large convective cell in the bulk is revealed by its thermal signature on the bottom and top plates. One frequency of the temperature oscillation is related to the velocity of this convective cell. We then obtain the typical temperature and velocity in the bulk versus the Rayleigh number, and compare them with similar results known for Pr∼1.We review two recent theoretical models, namely the mixing zone model of Castaing et al. (1989), and a model of the turbulent boundary layer by Shraiman & Siggia (1990). We discuss how these models fail at low Prandtl number, and propose modifications for this case. Specific scaling laws for fluids at low Prandtl number are then obtained, providing an interpretation of our experimental results in mercury, as well as extrapolations for other liquid metals.
306 citations
TL;DR: In this article, the authors present a systematic study of three potentially important effects: depth-dependent viscosity, an endothermic phase change, and bottom versus internal heating.
Abstract: Mantle convection is influenced simultaneously by a number of physical effects: brittle failure in the surface plates, strongly variable viscosity, mineral phase changes, and both internal heating (radioactivity) and bottom heating from the core. Here we present a systematic study of three potentially important effects: depth-dependent viscosity, an endothermic phase change, and bottom versus internal heating. We model three-dimensional spherical convection at Rayleigh Ra=108 thus approaching the dynamical regime of the mantle. An isoviscous, internally heated reference model displays point-like downwellings from the cold upper boundary layer, a blue spectrum of thermal heterogeneity, and small but rapid time variations in flow diagnostics. A modest factor 30 increase in lower mantle viscosity results in a planform dominated by long, linear downwellings, a red spectrum, and great temporal stability. Bottom heating has the predictable effect of adding a thermal boundary layer at the base of the mantle. We use a Clapeyron slope of γ=−4 MPa °K−1 for the 670 km phase transition, resulting in a phase buoyancy parameter of P=−0.112. This phase change causes upwellings and downwellings to pause in the transition zone but has little influence on the inherent time dependence of flow and only a modest reddening effect on the heterogeneity spectrum. Larger values of P result in stronger effects, but our choice of P is likely already too large to be representative of the mantle transition zone. Combinations of all three effects are remarkably predictable in terms of the single-effect models, and the effect of depth-dependent viscosity is found to be dominant.
271 citations
TL;DR: In this paper, the authors describe a series of laboratory experiments in which aqueous salt solutions were cooled and solidied from above, serving as model systems of metallic castings, magma chambers and sea ice.
Abstract: We describe a series of laboratory experiments in which aqueous salt solutions were cooled and solidied from above. These solutions serve as model systems of metallic castings, magma chambers and sea ice. As the solutions freeze they form a matrix of ice crystals and interstitial brine, called a mushy layer. The brine initially remains conned to the mushy layer. Convection of brine from the interior of the mushy layer begins abruptly once the depth of the layer exceeds a critical value. The principal path for brine expelled from the mushy layer is through ‘brine channels’, vertical channels of essentially zero solid fraction, which are commonly observed in sea ice and metallic castings. By varying the initial and boundary conditions in the experiments, we have been able to determine the parameters controlling the critical depth of the mushy layer. The results are consistent with the hypothesis that brine expulsion is initially determined by a critical Rayleigh number for the mushy layer. The convection of salty fluid out of the mushy layer allows additional solidication within it, which increases the solid fraction. We present the rst measurements of the temporal evolution of the solid fraction within a laboratory simulation of growing sea ice. We show how the additional growth of ice within the layer aects its rate of growth.
183 citations
TL;DR: In this article, the effect of surface radiation on the flow field, temperature distribution, and heat transfer is predicted, and it is shown that surface radiation significantly altered the temperature distribution and the flow patterns, especially at higher Rayleigh numbers.
Abstract: The interaction of natural convection with thermal radiation of gray surfaces in a square enclosure filled with air has been numerically investigated. The effect of radiation on the flow field, temperature distribution, and heat transfer is predicted. The result shows that surface radiation significantly altered the temperature distribution and the flow patterns, especially at higher Rayleigh numbers. The average convection Nusselt number increases with the increase of Ra. The presence of surface radiation can change the value of average convection Nusselt number, but only little variation can be observed with the increase of emissivity. The average radiative Nusselt number rises quickly with the increase of emissivity, and radiation heat transfer plays an important part in overall heat flux at larger emissivity. The correlation of entire average Nusselt number has also been discussed for evaluating heat transfer through the enclosure,
140 citations
TL;DR: In this article, the authors considered finite-amplitude convection in rotating spherical fluid shells for a variety of Prandtl numbers P and Rayleigh numbers Ra up to about 10 times the critical value.
Abstract: Finite-amplitude convection in rotating spherical fluid shells is considered for a variety of Prandtl numbers P and Rayleigh numbers Ra up to about 10 times the critical value. Convection at low Rayleigh numbers in the form of azimuthally periodic or weakly aperiodic drifting waves is characterized by relatively low heat transport, especially for P ≤ 1. The transition to strongly time-dependent convection leads to a rapid increase of the heat transport with increasing Rayleigh numbers. Onset of convection in the polar regions is delayed, but contributes a disproportionate fraction of the heat transport at high Rayleigh number. The differential rotation generated by convection, the distributions of helicity, and the role of asymmetry with respect to the equatorial plane are also studied.
127 citations
TL;DR: In this article, the authors used seafloor heat flow along seismic reflection profiles crossing a buried ridge on the eastern flank of the Juan de Fuca Ridge to infer hydrothermal heat-transport properties of the upper oceanic crust at this 3.5 Ma site.
118 citations
TL;DR: In this article, a numerical study was conducted to investigate steady state heat transfer and flow characteristics of natural convection in a vertical square enclosure when a temperature difference exists across an enclosure and, at the same time, a conducting body generates heat within the enclosure.
Abstract: A numerical study has been conducted to investigate steady state heat transfer and flow characteristics of natural convection in a vertical square enclosure when a temperature difference exists across an enclosure and, at the same time, a conducting body generates heat within the enclosure. Dimensionless governing equations indicate that the heat transfer and flow characteristics of this system are governed by the Rayleigh and Prandtl numbers, the area ratio, the conductivity ratio, and the temperature-difference ratio. Here the temperature-difference ratio is defined as the ratio of a temperature difference across the enclosure to that caused by the heat source. In the present study, the Rayleigh number ranges from 103 to 104, and the temperature-difference ratio from 0 to 50, while the Prandtl number, the area ratio, and the conductivity ratio are kept constant at 0.71, 0.25, 1, respectively. The analysis is performed by observing variations of streamlines, isotherms, heatlines, and the average Nusselt ...
111 citations
TL;DR: In this paper, the authors investigated large Rayleigh number (106−109) and large Prandtl number (102−103) thermal convection in glycerol in an aspect ration one cubic cell.
Abstract: We investigate large Rayleigh number (106–109) and large Prandtl number (102–103) thermal convection in glycerol in an aspect ration one cubic cell. The kinematic viscosity of the fluid strongly depends upon the temperature. The symmetry between the top and bottom boundary layers is thus broken, the so-called non-Boussinesq regime. In a previous paper Wu and Libchaber have proposed that in such a state the two thermal boundary layers adjust their length scales so that the mean hot and cold temperature fluctuations are equal in the center of the cell. We confirm this equality. A simplified two-dimensional model for the mean center temperature based on an equation for the thermal boundary layer is presented and compared with the experimental results. The conclusion is that the central temperature adjusts itself so that heat fluxes from boundaries are equal, temperature fluctuations at the center symmetrical, at a cost of very different temperature drops and Rayleigh number for each boundary.
102 citations
TL;DR: In this paper, the authors developed a model to study the mixed convection processes below a saline disposal basin located between a recharge and discharge zone and performed numerical simulations in cross-section using the 2-D density dependent model SUTRA (saturated-unsaturated transport).
TL;DR: In this paper, three-dimensional numerical convection calculations in a wide (8 × 8 × 1) cartesian box and in a spherical shell (ratio of inner to outer radius of 0.55, characteristic of terrestrial planets) both display two fundamental transitions as the viscosity contrast is progressively increased from unity to a factor of 105.
TL;DR: In this paper, the authors investigated linear and non-linear properties of thermal convection in rotating spherical shells of varying radius ratios and determined the range of validity of the simple "equatorial" approximation through a comparison with the more complete numerical analysis based on the Galerkin approximation.
TL;DR: In this paper, a progressive asymptotic approach procedure is presented for solving the steady-state Horton-Rogers-Lapwood problem in a fluid-saturated porous medium.
Abstract: In this paper, a progressive asymptotic approach procedure is presented for solving the steady-state Horton-Rogers-Lapwood problem in a fluid-saturated porous medium The Horton-Rogers-Lapwood problem possesses a bifurcation and, therefore, makes the direct use of conventional finite element methods difficult Even if the Rayleigh number is high enough to drive the occurrence of natural convection in a fluid-saturated porous medium, the conventional methods will often produce a trivial non-convective solution This difficulty can be overcome using the progressive asymptotic approach procedure associated with the finite element method The method considers a series of modified Horton-Rogers-Lapwood problems in which gravity is assumed to tilt a small angle away from vertical The main idea behind the progressive asymptotic approach procedure is that through solving a sequence of such modified problems with decreasing tilt, an accurate non-zero velocity solution to the Horton-Rogers-Lapwood problem can be obtained This solution provides a very good initial prediction for the solution to the original Horton-Rogers-Lapwood problem so that the non-zero velocity solution can be successfully obtained when the tilted angle is set to zero Comparison of numerical solutions with analytical ones to a benchmark problem of any rectangular geometry has demonstrated the usefulness of the present progressive asymptotic approach procedure Finally, the procedure has been used to investigate the effect of basin shapes on natural convection of pore-fluid in a porous medium (C) 1997 by John Wiley & Sons, Ltd
TL;DR: In this paper, the authors present an experimental and theoretical study that reveals that the brine expelled by growing sea ice initially remains trapped within the crystalline interstices but that drainage into the underlying water is ultimately triggered abruptly due to the onset of buoyancy-driven convection.
Abstract: Sea ice forms as a porous matrix consisting of pure ice crystals in equilibrium with brine. We present an experimental and theoretical study that reveals that the brine expelled by growing sea ice initially remains trapped within the crystalline interstices but that drainage into the underlying water is ultimately triggered abruptly due to the onset of buoyancy-driven convection. The onset is quantified by a critical porous-medium Rayleigh number, and we present an empirical marginal stability diagram that defines the transition.
TL;DR: In this article, the horizontal scale of rotating convection with rigid boundary conditions is studied, and the experimental results agree fairly well with the estimated scale, which depends on the ratio between the thicknesses of the Ekman layer and the thermal boundary layer, and does not depend monotonically on the Rayleigh number.
Abstract: The horizontal scale of rotating convection with rigid boundary conditions is studied. The range of Rayleigh number concerned is moderate, i.e. large enough to induce a finite-amplitude convection but small enough so that the geostrophic processes are significant.On considering an experimental law of the Nusselt number and some constraints of elemental geostrophic processes, the horizontal scale of the convection can be estimated. This estimation strongly depends on the ratio between the thicknesses of the Ekman layer and the thermal boundary layer, and does not depend monotonically on the Rayleigh number. This dependency is compatible with the experimental results of Rossby (1969).The estimated horizontal scale was checked by laboratory experiments. The horizontal temperature distribution was visualized by thermal liquid-crystal capsules dispersed in the working fluid. The horizontal scale was measured by counting vortices. The experimental results agree fairly well with the estimated scale.
TL;DR: In this article, a numerical study of combined heat and mass transfer by natural convection adjacent to vertical surfaces situated in fluid-saturated porous media is reported, where the structure of the flow, temperature and concentration fields are governed by complex interactions among the diffusion rates and the buoyancy ratio.
TL;DR: In this paper, the transition from the onset of convection to fully developed turbulence of a Rayleigh-Benard flow, in a low-aspect-ratio cell and in mercury, is studied through three-dimensional numerical simulation of the Navier-Stokes equations.
Abstract: The transitions from the onset of convection to fully developed turbulence of a Rayleigh–Benard flow, in a low-aspect-ratio cell and in mercury, are studied through three-dimensional numerical simulation of the Navier–Stokes equations. The calculation of the growth rate of the azimuthal energy modes permitted the accurate determination of the critical Rayleigh number for the establishment of the convective regime (Rac=3750) which is in good agreement with analytical and other numerical results. Increasing the Rayleigh number, the flow remained steady up to Ra≃2.11×104 when an oscillatory instability was observed. Further increases in the Rayleigh produced a chaotic state through the period doubling mechanism and finally the turbulent state was achieved. It is shown that for Ra⩾Rac the mean flow consists of a large-scale convective cell which persists in the whole range of studied Rayleigh numbers (Ra⩽106). The dependence of the Nusselt number over the Rayleigh number is also analyzed and, for Ra⩾3.75×104,...
TL;DR: In this paper, the effects of free convection currents with one relaxation time on the flow of a viscoelastic conduction fluid through a porous medium, which is bounded by a vertical plane surface, have been studied.
Abstract: Effects of free convection currents with one relaxation time on the flow of a viscoelastic conduction fluid through a porous medium, which is bounded by a vertical plane surface, have been studied. The state space approach developed in Ezzat (Can. J. Phys, Vol. 72, p. 311, 1994; J. Appl. Math. Comput., Vol. 64, p. 1, 1994) is adopted for the solution of one-dimensional problems in magnetohydrodynamic free convection flow with thermal relaxation time. The resulting formulation together with the Laplace transform technique is applied to a variety of problems. The solutions to a problem of an electrically conducting visocelastic fluid in the presence of hydromagnetic free convection flow and to a problem of the flow between two parallel fixed plates are obtained. A discussion of the effects of cooling and heating on a viscoelastic conducting fluid is given. Numerical results are illustrated graphically for both problems considered.
TL;DR: In this article, the effect of surface undulations on the natural convection heat transfer from an isothermal surface in a Darcian fluid-saturated porous enclosure has been numerically analyzed using the finite element method on a graded nonuniform mesh system.
Abstract: The effect of surface undulations on the natural convection heat transfer from an isothermal surface in a Darcian fluid-saturated porous enclosure has been numerically analyzed using the finite element method on a graded nonuniform mesh system. The flow-driving Rayleigh number Ra together with the geometrical parameters of wave amplitude a, wave phase φ, and the number of waves N considered in the horizontal dimension of the cavity are found to influence the flow and heat transfer process in the enclosure. For Ra around 50 and above, the phenomenon of flow separation and reattachment is noticed on the waits of the enclosure. A periodic shift,n the reattachment point from the bottom wall to the adjacent walls in the clockwise direction, leading to the manifestation of cycles of unicellular and bicellular clockwise and counterclockwise flows, is observed, with the phase varying between 0° and 350°. The counterflow in the secondary circulation zone is intensified with the increase in the value of Ra...
TL;DR: In this article, the authors present a review of the available turbulence modelling techniques, with particular emphasis on selecting models capable of treating the mechanisms of turbulence that are relevant to high Rayleigh number natural convection.
TL;DR: In this paper, the stability of a horizontal fluid and fluid-saturated porous layer heated from below is examined for the case of a time-dependent buoyancy force generated by gravity modulation.
TL;DR: In this article, a finite element method was used to solve conjugate conduction with natural convection in a square enclosure, where the set of governing equations have been reframed and put into a form that can easily fit into the Galerkin formulation.
TL;DR: In this article, the authors used an axisymmetric spherical shell model to simulate the exothermic olivine-spinel (α-β, β-γ) phase transition in the Martian mantle and found that the latent heat release causes the mantle temperature to increase across each transition by about 50 K and produces a hot lower mantle and a liquid core.
TL;DR: In this article, the effects of the Prandtl number on heat transfer in volumetrically heated liquid pools with Rayleigh numbers up to 1012 have been investigated, with particular emphasis on the analysis of Pr number effects.
TL;DR: In this article, the effect of inclination on laminar natural convection in a square cavity is studied numerically for inclination angles ranging from 40° to 160°, Rayleigh numbers between 103 and 106 and Prandtl numbers from 0.02 to 4,000.
Abstract: The effect of inclination on laminar natural convection in a square cavity is studied numerically for inclination angles ranging from 40° to 160°, Rayleigh numbers between 103 and 106 and Prandtl numbers from 0.02 to 4,000. Contours of stream functions and temperature are presented in order to provide a new insight and a better understanding of the flow and heat transfer characteristics in a square cavity. Finds computed local and mean Nusselt numbers at the hot wall in satisfactory agreement with experimental and other numerical results. Finds the mean heat flux at the hot wall as well as the distribution of the local heat flux along the heated wall are found to depend on the inclination angle, the Rayleigh number and the Prandtl number. Such a dependence is significant for angles greater than 90°, while for smaller angles the effect of the inclination angle on the Nusselt‐Rayleigh correlation is weaker.
TL;DR: In this paper, an experimental investigation of Rayleigh-Benard convection in binary-gas mixtures is presented, in which the necessary thermodynamic and transport properties for six mixtures are determined by a combination of data from the literature, molecular-theory calculations, and thermal conductivity measurements.
Abstract: We present an experimental investigation of Rayleigh-Benard convection in binary-gas mixtures. In order to interpret the results quantitatively, we determined the necessary thermodynamic and transport properties for six mixtures ~He-CO2, He-SF 6, He-Xe, Ne-Ar, Ar-CO2, and H 2-Xe! by a combination of data from the literature, molecular-theory calculations, and thermal-conductivity measurements. All six mixtures have positive separa- tion ratios C. The Lewis numberL ~the ratio of the mass to the thermal diffusivity! is of O(1), in contrast to liquid mixtures whereL5O(10 22 ). An important feature of the gas mixtures is that their Prandtl number ~the ratio of the kinematic viscosity to the thermal diffusivity ! can be lower than those of the two pure components. We discuss the physical reason for this and show that the minimum Prandtl number reached by using binary- gas mixtures is about 0.16. The critical temperature difference DTc for the onset of convection is determined from measurements of the Nusselt numberN ~the effective thermal conductivity! and from the contrast of shadowgraph images as a function of DT. The results agree well with the prediction of linear stability analysis. In contrast to convection in binary-liquid mixtures with C.0, N for the gas mixtures increases significantly with e(DT/DTc21 as soon as the convection starts at the Soret onset and is qualitatively similar to the Nusselt number of pure fluids. However, the critical Rayleigh number Rc is lower than the value Rc051708 of pure fluids. The pattern at onset in the gas mixtures initially consists of parallel straight rolls, in contrast to binary-liquid mixtures where the pattern consists of squares. Based on the gas-mixture properties, we find that the Dufour effect ~the reciprocal process of the Soret effect! is relatively weak. The slope dN/de ofN at onset is found to be consistent with that predicted by an eight-mode Galerkin truncation. @S1063-651X~97!04206-2#
TL;DR: In this article, a theory of nonlinear evolution and secondary instabilities in surface-tension-driven convection in a two-layer liquid-gas system with a deformable interface, heated from below is presented.
Abstract: The paper presents a theory of nonlinear evolution and secondary instabilities in Marangoni (surface-tension-driven) convection in a two-layer liquid–gas system with a deformable interface, heated from below. The theory takes into account the motion and convective heat transfer both in the liquid and in the gas layers. A system of nonlinear evolution equations is derived that describes a general case of slow long-scale evolution of a short-scale hexagonal Marangoni convection pattern near the onset of convection, coupled with a long-scale deformational Marangoni instability. Two cases are considered: (i) when interfacial deformations are negligible; and (ii) when they lead to a specific secondary instability of the hexagonal convection.In case (i), the extent of the subcritical region of the hexagonal Marangoni convection, the type of the hexagonal convection cells, selection of convection patterns – hexagons, rolls and squares – and transitions between them are studied, and the effect of convection in the gas phase is also investigated. Theoretical predictions are compared with experimental observations.In case (ii), the interaction between the short-scale hexagonal convection and the long-scale deformational instability, when both modes of Marangoni convection are excited, is studied. It is shown that the short-scale convection suppresses the deformational instability. The latter can appear as a secondary long-scale instability of the short-scale hexagonal convection pattern. This secondary instability is shown to be either monotonic or oscillatory, the latter leading to the excitation of deformational waves, propagating along the short-scale hexagonal convection pattern and modulating its amplitude.
TL;DR: In this paper, the authors studied the convection heat transfer in inclined rectangular enclosures with perfectly conducting fins attached to the heated wall and found that the heat transfer through the cover is considerably affected by the presence of the fins.
Abstract: The natural convection heat transfer in inclined rectangular enclosures with perfectly conducting fins attached to the heated wall is numerically studied. The parameters governing this problem are the Rayleigh number (102≤Ra≤2×105), the aspect ratio of the enclosures (2.5≤A=H′/L′≤∞), the dimensionless lengths of the partitions (0≤B=l′/L′≤1), the aspect ratio of micro-cavities (A≤C=h′/L′≤0.33), the inclination angle (0≤φ≤60∘) and the Prandtl number (Pr=0.72). The results indicate that the heat transfer through the cover is considerably affected by the presence of the fins. At low Rayleigh numbers, the heat transfer regime is dominated by conduction. When B≈0.75 and C≈0.33, the heat transfer through the cold wall decreases considerably. This trend is enhanced when the enclosure is inclined. Useful engineering correlations are derived for practical applications.
TL;DR: In this paper, a linear proportional control algorithm was proposed to stabilize the unstable no-motion state in a moderate aspect ratio one-dimensional Rayleigh-Benard convection experiment.
Abstract: We report on stabilizing the unstable no-motion state in a moderate aspect ratio one-dimensional Rayleigh–Benard convection experiment. A linear proportional control algorithm uses shadowgraphic convection images to determine heat flux perturbations which are applied to the lower boundary by a network of local heaters. We show that simple linear control stabilizes the otherwise unstable no-motion (conduction) state over a substantial range of supercritical Rayleigh numbers.