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Showing papers on "Rayleigh number published in 1981"


Journal ArticleDOI
TL;DR: In a horizontal layer of fluid heated from below and cooled from above, cellular convection with horizontal length scale comparable to the layer depth occurs for small enough values of the Rayleigh number.
Abstract: In a horizontal layer of fluid heated from below and cooled from above, cellular convection with horizontal length scale comparable to the layer depth occurs for small enough values of the Rayleigh number. As the Rayleigh number is increased, cellular flow disappears and is replaced by a random array of transient plumes. Upon further increase, these plumes drift in one direction near the bottom and in the opposite direction near the top of the layer with the axes of plumes tilted in such a way that horizontal momentum is transported upward via the Reynolds stress. With the onset of this large-scale flow, the largest scale of motion has increased from that comparable to the layer depth to a scale comparable to the layer width. The conditions for occurrence and determination of the direction of this large-scale circulation are described.

337 citations


Journal ArticleDOI
TL;DR: In this article, the effect of the partitions on the heat transfer across the enclosure is also studied and correlations for the Nusselt number as a function of RaL and partition length are generated for both conducting and nonconducting partition materials.
Abstract: / Heat transfer by natural convection in a two-dimensional rectangular enclosure fitted with partial vertical divisions is investigated experimentally. The horizontal walls of the enclosure are adiabatic while the vertical walls are maintained at different temperatures. The experiments are carrir8 out with water, n 3.5, for Rayleigh numbers in the range, 2.3 X 10 < RaL < 1.1 x 10 , and an aspect ratio, A = H/L = 1/2. The effect of the partial-vertical divisions on the fluid flow and temperature fields is investigated by dye-injection flow visualization and by thermocouple probes, respectively. The effect of the partitions on the heat transfer across the enclosure is also studied and correlations for the Nusselt number as a function of RaL and partition length are generated for both conducting and non-conducting partition materials. Partial divisions are found to have a significant effect on the heat transfer, especially when the divisions are adiabatic. The results also indicate that the partial divisions may have a stabilizing effect on the laminar-transitional flow on the heated vertical walls of the enclosure.

137 citations


Journal ArticleDOI
TL;DR: In this paper, the effects of Gr (Grashof number), Gm (modified Grashof numbers) and permeability K of the porous medium on the velocity and rate of heat transfer are discussed when the surface is subjected to a constant suction velocity.

119 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection, and showed that period-doubling is a characteristic feature of double-diffusive convection.
Abstract: We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently small and the solutal Rayleigh number, RS, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, RT, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of RS the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of RS a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of RT. The chaotic solutions persist as RT is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that period-doubling, followed by the appearance of a strange attractor, is a characteristic feature of double-diffusive convection.

109 citations



Journal ArticleDOI
TL;DR: In this article, steady solutions in the form of two-dimensional rolls are obtained numerically for convection in a horizontal layer of a low-Prandtl-number fluid heated from below.
Abstract: Steady solutions in the form of two-dimensional rolls are obtained numerically for convection in a horizontal layer of a low-Prandtl-number fluid heated from below. Prandtl numbers in the range 0·001 [les ] P [les ] 0·71 are investigated for Rayleigh numbers between the critical value, R = 1708, and R = 20,000 in the case of rigid boundaries. The calculations reveal that the convective heat transport is relatively independent of the Prandtl number at Rayleigh numbers greater than a finite critical value R2 of the order of 5 × 103. At R = 10,000 the convective heat transport varies by only about 30% for Prandtl numbers in the range investigated. As the Rayleigh number is increased above the critical value R2, the streamlines of the convection flow become circular, independent of the horizontal wavelength as long as the latter is larger than or about equal to twice the height of the layer.

101 citations


Journal ArticleDOI
TL;DR: In this paper, nonlinear two-dimensional magnetoconvection in a Boussinesq fluid has been studied in a series of numerical experiments with values of the Chandrasekhar number Q ≤ 4000 and the ratio ζ of the magnetic to the thermal diffusivity in the range 1 ≥ ζ ≥ 0·025.
Abstract: Nonlinear two-dimensional magnetoconvection in a Boussinesq fluid has been studied in a series of numerical experiments with values of the Chandrasekhar number Q ≤ 4000 and the ratio ζ of the magnetic to the thermal diffusivity in the range 1 ≥ ζ ≥ 0·025. If the imposed field is strong enough, convection sets in as overstable oscillations which give way to steady convection as the Rayleigh number R is increased. In the dynamical regime that follows, magnetic flux is concentrated into sheets at the sides of the cells, from which the motion is excluded.

101 citations



Journal ArticleDOI
TL;DR: In this paper, the behavior of an extra roll extending into an otherwise regular convection pattern is studied as a function of Rayleigh number, Prandtl number, $P$, and wavelength, by means of a fully resolved numerical simulation of the Boussinesq equations with free-slip boundary conditions.
Abstract: The behavior of an extra roll extending into an otherwise regular convection pattern is studied as a function of Rayleigh number, Prandtl number, $P$, and wavelength, by means of a fully resolved numerical simulation of the Boussinesq equations with free-slip boundary conditions. For reduced Rayleigh numbers of order one or less and $P\ensuremath{\gtrsim}40$, numerical simulations of the lowest-order amplitude equations reproduce the Boussinesq results semiquantitatively. In particular, we find that when this class of defects is stable, they move with constant velocity $v$, parallel to the roll axis and give rise to a slow modulation of the roll pattern of the form $f(x,y\ensuremath{-}vt)$. Both $f$ and $v$ have been calculated analytically within a linearized theory. The envelope function $f$ depends in an essential way on $v$ such that the limit $v\ensuremath{\rightarrow}0$ cannot be sensibly taken.

100 citations


Book ChapterDOI
01 Jan 1981

98 citations


Journal ArticleDOI
TL;DR: In this article, the effects of G (Grashof number) and K (permeability parameter) on the velocity and the rate of heat transfer of two-dimensional free convection flow through a porous medium bounded by a vertical infinite surface is considered.
Abstract: Steady two-dimensional free convection flow through a porous medium bounded by a vertical infinite surface is considered. Expressions for the velocity, temperature and the rate of heat transfer are obtained. Effects of G (Grashof number) and K (permeability parameter) on the velocity and the rate of heat transfer are discussed.

Journal ArticleDOI
TL;DR: In this article, a simplified model for two-dimensional convection in the presence of an imposed magnetic field is described, which is exact to second order in the amplitude of the motion and appears to be qualitatively correct for larger amplitudes.
Abstract: Two-dimensional convection in a Boussinesq ffuid in the presence of an imposed magnetic field is described in terms of a simplified model, which is exact to second order in the amplitude of the motion and appears to be qualitatively correct for larger amplitudes. If the ratio of the magnetic diffusivity to the thermal diffusivity is SUEciently small and the imposed magnetic field is sufficiently large, convection sets in when r = r(O) as overstable oscillations, which grow in amplitude as the normalized Rayleigh number r is increased. There is also a branch of steady solutions that bifurcates from the static equilibrium at r = r"' > do) and stable steady solutions exist for r > rmin. For certain choices of parameters subcritical steady convection, with rmin < de), is found and the oscillatory branch ends on the unstable portion of the steady branch, where the period of the oscillations becomes infinite. In some circumstances there may be a bifurcation from symmetrical to asymmetrical oscillations, followed by a sequence of bifurcations at each of which the period doubles. Other choices of parameters allow only supercritical convection with r increasing monotonically on the steady branch; if convection first appears as overstable oscillations the steady branch is then unstable for rce) < r < rmin and there is a Hopf bifurcation at r = rmin. This complicated pattern of behaviour is consistent with the results of numerical experiments on the full two-dimensional problem.


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the heat transported by convection at slightly supercritical Rayleigh numbers and showed that the slope of the Nusselt-number graph at the critical point is nearly independent of cell width.
Abstract: Previous work by the present authors on the onset of convection in a layered porous medium heated from below is extended to an investigation of the heat transported by convection at slightly supercritical Rayleigh numbers.The two-dimensional convection patterns and associated values of the critical Rayleigh number, cell width and slope of the Nusselt-number graph are calculated for two- and three-layer configurations over a wide range of layer depth and permeability ratios. The results show that the commonly studied problem of a homogeneous layer bounded above and below by impermeable boundaries is a special case, in that the slope of the Nusselt-number graph at the critical point is nearly independent of cell width. For a homogeneous layer with a permeable upper boundary, and for multi-layered systems, the slope of this graph depends strongly on cell width.

Journal ArticleDOI
TL;DR: In this paper, the authors reviewed the possible flows in a pure melt with respect to the ratio GrRe2, where Gr is the Grashof number due to the temperature difference between the crystal and crucible and Re is the Reynolds number of the rotating crystal.

Journal ArticleDOI
TL;DR: In this paper, the problem of steady film condensation outside a wedge or a cone embedded in a porous medium filled with a dry saturated vapor is investigated, where the condensate and the vapor are separated by a distinct boundary with no two-phase zone in between.

Book ChapterDOI
01 Jan 1981
TL;DR: In this paper, an anelastic approximation for stellar convection is proposed, which filters out sound waves but permits the investigation of the effects of large density stratifications in slightly superadiabatic stellar envelopes.
Abstract: Giant cell stellar convection is modeled by solving the fluid equations for a compressible, rotating, spherical, fluid shell. A large part of the motivation is to understand the maintenance of the two major ingredients in solar dynamo theory, that is helicity and differential rotation. An anelastic approximation filters out sound waves but permits the investigation of the effects of large density stratifications in slightly superadiabatic stellar envelopes. Various rotation rates, convection zone depths, density stratifications, boundary conditions, viscosities, and conductivities are considered. The results of first order numerical calculations for the onset of convection are discussed with emphasis on the structure of the most unstable modes. Left (right) handed helical motion dominates in the northern (southern) hemisphere. Also, as the stratification increases, the horizontal dimension of the most unstable modes decreases, the prograde phase velocity increases, and the buoyancy force does more negative work in the upper part of the convection zone. Differential rotation is maintained by the transport of longitudinal momentum and by the coriolis forces acting on the meridional circulation. Second order numerical calculations provide profiles of the differential rotation and meridional circulation induced by the first order perturbations. Results of these calculations for the most unstable modes show that either equatorial acceleration, as observed on the sun, or equatorial deceleration can be maintained depending on the rotation rate, density stratification, viscosity, and conductivity. Small viscous diffusion relative to thermal diffusion is required for equatorial acceleration in rapidly rotating, highly stratified convection zones.

Journal ArticleDOI
TL;DR: In this paper, an experimental study of buoyancy-induced motion and heat transfer in a horizontal rectangular cavity with the two vertical ends at different temperatures and the long horizontal walls adiabatic is presented.
Abstract: The paper summarizes results from an experimental study of buoyancy-induced motion and heat transfer in a horizontal rectangular cavity with the two vertical ends at different temperatures and the long horizontal walls adiabatic. The cavity height/ length ratio is A = 0.0625. The high-Rayleigh-number range reported on in this paper, 2 x 108 < Ra < 2 x 109, has not been studied before. It is shown that, contrary to lower-Rayleigh-number behaviour known previously, the core flow structure is nonparallel and is dominated by horizontal intrusions flowing along each of the two insulated horizontal walls of the enclosure. The fluid embraced by the two horizontal jets is practically stagnant and thermally stratified. Flow visualization experiments suggest that adjacent to the two horizontal jets two secondary flat cells are formed by the baroclinic pressure field in an analogous way to what is observed in intrusions in a stratified fluid. Nusselt-number-Rayleigh-number results for the overall end-to-end heat transfer in the horizontal direction are reported and compared with previous experimental and theoretical results available for lower Rayleigh numbers. It is shown also that the transition from a parallel core structure to one dominated by intrusion layers is governed by the parameter RaiA, with RatA < 1 aa necessary condition for a parallel core flow.

Journal ArticleDOI
TL;DR: In this paper, the stability of doubly diffusive fluid is investigated and it is shown that a non-rotating layer can be destabilized by rotation, a rotating layer can also be destabilised by the addition of a bottom-heavy solute gradient, and under some conditions, three thermal Rayleigh numbers are required to specify linear stability criteria.
Abstract: The stability of a rotating doubly diffusive fluid is considered. It is shown that (1) a non-rotating layer can be destabilized by rotation, (2) a rotating layer can be destabilized by the addition of a bottom-heavy solute gradient, and (3) under some conditions, three thermal Rayleigh numbers are required to specify linear stability criteria. Numerical results are presented on the basis of which the explanation by Acheson (1979) of the second of these three anomalies can be assessed, and Acheson's explanation is adapted to the two other anomalies.

Journal ArticleDOI
TL;DR: In this article, it was shown that, for any value of the salt Rayleigh number R S, finite-amplitude convection can occur at values of the Ray-leigh number T much less than that necessary for infinitesimal oscillations, provided only that T is sufficiently small.
Abstract: Steady convective motions in a Boussinesq fluid with an unstable thermal and stable salinity stratification are investigated in the case that the ratio of diffusivities τ = κ S /κ T [Lt ] 1. Using perturbation theory, it is shown that, for any value of the salt Rayleigh number R S , finite-amplitude convection can occur at values of the Ray-leigh number R T much less than that necessary for infinitesimal oscillations, provided only that T is sufficiently small. A simple qualitative argument is used to show how R min , the minimum value of R T for steady convection, varies with R S , and it is shown that the analytical results of the present paper form a natural complement to the numerical ones of Huppert & Moore (1976). Results are presented both for stress-free and for rigid boundaries, and applicability of the method to other related problems is suggested.

Journal ArticleDOI
TL;DR: In this paper, a numerical study of steady, natural convection in a fluid-saturated, horizontal, porous layer subjected to an end-to-end temperature difference was performed using a finite element computer program based on the Galerkin form of the finite element method.
Abstract: A numerical study of steady, natural convection in a fluid-saturated, horizontal, porous layer subjected to an end-to-end temperature difference is reported. The analysis is performed using a finite element computer program based on the Galerkin form of the finite element method. Heat transfer rates are predicted for aspect ratios ranging from 0.1 to 0.5 and Rayleigh numbers in the range 25 to 200. Representative plots of temperature and velocity fields are presented. Comparisons are made with an approximate analytical solution and regions of validity are identified for the analytical solution.

Journal ArticleDOI
TL;DR: In this article, the authors investigated two-dimensional magnetoconvection for a fixed Rayleigh number of 104, with the ratio ζ of the magnetic to the thermal diffusivity in the range 0·4 ≥ ζ ≥ 0·05.
Abstract: Nonlinear, two-dimensional magnetoconvection has been investigated numerically for a fixed Rayleigh number of 104, with the ratio ζ of the magnetic to the thermal diffusivity in the range 0·4 ≥ ζ ≥ 0·05. As the Chandrasekhar number Q is decreased, convection first sets in as overstable oscillations, which are succeeded by steady convection with dynamically active flux sheets and, eventually, with kinematically concentrated fields. In the dynamical regime spatially asymmetrical convection, with most of the flux on one side of the cell, is preferred. As Q increases, these asymmetrical solutions become time-dependent, with oscillations about the steady state which develop into large-scale oscillations with reversals of the flow. Although linear theory predicts that narrow cells should be most unstable, the nonlinear results show that steady convection occurs most easily in cells that are roughly twice as wide as they are deep.


Journal ArticleDOI
TL;DR: Luijkx and Platten as mentioned in this paper studied the onset of convection in an infinite channel of rectangular cross section and heated from below; the four boundaries are supposed to be rigid and perfectly heat-insulating.
Abstract: We study the onset of convection in an infinite channel of rectangular cross section and heated from below; the four boundaries are supposed to be rigid. Three-dimensional transverse rolls are preferred to longitudinal rolls and to finite transverse rolls. We also describe the shape of the convective motion at the critical point. Introduction The study of the onset of convection inside a completely confined region began with Davis in 1967, [1]; this author, considering a fluid maintained in a rectangular parallelepiped heated from below, performed a linear analysis and predicted for a critical thermal gradient the appearance of \"convective finite rolls (rolls with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter sides of the box\". This earlier work, already criticized by Catton [2], regarding the validity of Davis' trial functions, was severely corrected by Davies-Jones [3], who demonstrated that the Davis' finite rolls could never be a solution of the equations that describe the movement of the fluid: the true structure at the onset of convection is purely threedimensional (no component of the fluid velocity vanishes identically). Yet, DaviesJones showed that, although the Davis' finite rolls could not exist, they represent a good approximation of the true, three-dimensional solution. To show the validity of this latest assertion, he calculated tKe stability of the movement of a fluid confined in an infinite channel of rectangular cross section, heated from below, and maintained between two \"free\" horizontal boundaries, which allows very simple — and exact — solutions of the equations but does not permit the comparison with experiments. The aim of this paper is a complete inquiry of the onset of convection in a \"reasonable\" apparatus (i.e. allowing comparison with experiments) — with rigid horizontal boundaries — and especially the investigation of the differences between the \"finite rolls\" approach and the real, three-dimensional convective structure. 0340-0204/81/0006-0141 $02.00 © Copyright by Walter de Gruyter & Co. Berlin New York 142 J. M. Luijkx, J. K. Platten 1. Basic equations A fluid is confined in an infinite channel of rectangular cross section (see Fig. 1). The horizontal boundaries (z = ± a) supposed to be rigid and perfectly heat-conducting are maintained at different temperatures. The lateral boundaries (y = ± b) are also rigid and perfectly heat-insulating. Fig. 1: System of coordinates. The boundary conditions are: = v(z=.±a) = and dy y=± = 0. (1) (2)


Journal ArticleDOI
TL;DR: In this article, a large roll with axis parallel to the cold wall was observed in a tank with two isothermal heat sinks: the top plate and one of the sidewalls, and the circulation in the large roll remains two-dimensional for Rayleigh numbers up to about 3 × 104.
Abstract: Convection experiments are carried out in a tank with two isothermal heat sinks: The top plate and one of the sidewalls. Heating is supplied either by an isothermal bottom plate or by generation within the fluid. This situation is similar to that of the earth's subcontinental mantle in the presence of a neighboring subducting oceanic lithosphere. There, the mantle material loses heat not only to the base of the continental lithosphere but also to the cold dipping oceanic slab. The thermal structure of the convective fluid is observed by a variety of techniques, including differential interferometry and strioscopy. Two parameters characterize the observations: the usual ‘vertical’ Rayleigh number and a newly defined ‘lateral’ Rayleigh number. The lateral cooling induces a large roll with axis parallel to the cold wall. The variation of its width relative to the values of the two Rayleigh numbers has been determined. An application to the earth's upper mantle would predict rolls 5 times wider than high. The circulation in the large roll remains two-dimensional for Rayleigh numbers up to about 3 × 104. Beyond this value, boundary layer instabilities are observed within the persisting large roll. Their period has been determined. The interferometric method is shown to be very useful for visualizing other time-dependent processes such as the growth of the induced large rolls. This growth is rapid enough to allow us to propose the existence of such large rolls in the earth and argue that their action could have led to continental break-up.

Journal ArticleDOI
TL;DR: In this paper, an approximate solution of two-dimensional convection in the limit of low Prandtl number is presented in which the buoyancy force is balanced by the inertial terms.
Abstract: An approximate solution of two-dimensional convection in the limit of low Prandtl number is presented in which the buoyancy force is balanced by the inertial terms. The results indicate that inertial convection becomes possible when the Rayleigh number exceeds a critical value of about 7 × 103. Beyond this value the velocity and temperature fields become independent of the Prandtl number except in thin boundary layers. The convective heat transport approaches the law Nu = 0·175 R¼ for the Nusselt number Nu. These results are in reasonably close agreement with the numerical results described in the preceding paper by Clever & Busse (1980).

Journal ArticleDOI
TL;DR: In this paper, the authors used the mantle Rayleigh number (R.P.B) to estimate the amount of heat lost from the earth's interior to the decay of radioactive isotopes and the cooling of the earth.

Journal ArticleDOI
Philip Marcus1
TL;DR: In this article, the Galerkin equations for convection in a sphere are examined to determine which physical processes are neglected by the severe truncation of the equations of motion, and it is demonstrated that the gross features of the flow are affected by truncation in the horizontal direction.
Abstract: The Galerkin equations for convection in a sphere are examined to determine which physical processes are neglected by the severe truncation of the equations of motion. It is demonstrated that the gross features of the flow are affected by truncation in the horizontal direction, with all of the models considered being well resolved in the vertical direction. One of the effects of truncation is to enhance the high-wave number end of the kinetic energy and thermal variance spectra. The examples cited indicate that as long as the kinetic energy spectrum decreases with wave number, a truncation gives a qualitatively correct solution. Conclusions are tested by calculating solutions to the equations of motion for several values of the Rayleigh number and the limit of horizontal spatial resolution.

Journal ArticleDOI
TL;DR: In this paper, the effects of spherical geometry, density interfaces, heat source distribution, and cell size on the surface velocities of the isoviscous spherical mantle convection were studied.
Abstract: Results of a similarity theory for spherical mantle convection are presented. The single-mode mean field equations are analyzed for convection which is so vigorous that temperature disturbances become localized in thin thermal boundary layers. Our purpose is to study effects of spherical geometry, density interfaces, heat source distribution, and cell size. Steady state solutions are found for isoviscous spherical shells in which the field of motion is spatially periodic in a single spherical harmonic degree. Calculations are carried out over the range 2 ≤ l ≤ 40 and for various fractions of internal versus base heating. Three configurations are examined: (1) convection in a single layer of cells extending through the whole mantle, (2) convection in two layers, separated by a density interface at 670-km depth, and (3) convection in a single layer terminating at 670 km. Results of these calculations are used to give estimates of surface horizontal velocities in terms of the heat loss, viscosity stratification, amount of internal heating, and depth of circulation. The surface velocity is most strongly affected by the thickness of the convecting shell. Deep mantle convection can achieve surface velocities which agree with observed plate speeds, while convection restricted to the upper mantle does not, at least on the scale of the major plates. The temperature distribution is strongly affected by the spherical geometry and by the presence of density interfaces. The principal difference between convection in one and two layers is that the latter produces a ‘hot’ lower mantle, while the former produces a ‘warm’ one.