Showing papers on "Rayleigh number published in 1985"

Layered convection induced by phase transitions

Abstract: We report a systematic study on the conditions under which an endothermic phase transition can enforce layered convection. Two-dimensional numerical calculations of convection in a domain containing a divariant phase change were performed in the framework of the “extended Boussinesq approximation,” i.e., considering the effects of adiabatic gradient, latent heat, and frictional heating in the energy equation. We find that the critical value of the negative Clapeyron slope, which must be surpassed in order to induce layered convection, decreases in magnitude with increasing Rayleigh number Ra in the range 104 ≤ Ra ≤ 2×106. Near the critical Clapeyron slope, vacillations between double- and single-layer convection or strongly leaking double-layer convection are possible. The breakdown into layers is influenced very little by the latent heat release but depends solely on the phase boundary deflection caused by lateral temperature differences. The value of the critical Clapeyron slope also seems little affected by the width of the transition zone or by its depth. A possible superplastic rheology within the transition zone would tend to favor layered convection. Scaling the model results to the 670-km discontinuity in the earth's mantle as a possible endothermic phase boundary, we estimate the critical Clapeyron slope to be in the range of −4 to −8 MPa/K (−40 to −80 bar/K). The possibility that the spinel → perovskite + periclase transition is within this range appears to be remote but certainly cannot be neglected.

The Formation of Freckles in Binary Alloys

Abstract: This paper presents a synopsis of some recent work, still in progress, aimed at elucidating a quantitative explanation of the processes by which flow chimneys form when certain types of alloys are directionally solidified. If (for example) light fluid is released at the liquid-solid "mushy" (dendrite) zone, and cooling is from below, then the intermediate fluid flow undergoes convection through the porous dendrite mass. This can lead to an "instability" of the form of the mushy zone, such that upwelling light fluid flows preferentially in channels within the dendrite mass. What we seek to develop here, is a mathematical basis by which this phenomenon may be properly understood. Accordingly, a mathematical model is developed, simplified, and partially analysed, and as a result we are able to make one specific prediction concerning a criterion for the onset of convection and freckling. This prediction is equivalent to the classical Rayleigh number condition for convective instability.

Natural convection in porous media

Abstract: This book presents the papers given at a conference on free convection in porous materials. Topics considered at the conference included heat transfer, nonlinear temperature profiles and magnetic fields, boundary conditions, concentrated heat sources in stratified porous media, free convective flow in a cavity, heat flux, laminar mixed convection flow, and the onset of convection in a porous medium with internal heat generation and downward flow.

Traveling waves and chaos in convection in binary fluid mixtures.

Abstract: Rayleigh-B\'enard convection is studied in alcohol-water mixtures in which the diffusion of concentration opposes convection via the Soret effect Near onset, the convective rolls are found to move continuously as traveling waves, in contrast to the stationary roll patterns observed in homogeneous fluids Dependent upon the temperature difference across the fluid layer (ie, Rayleigh number), these traveling-wave states are either periodic or chaotic At larger Rayleigh numbers, time-independent flow is observed which is the same as that expected for the homogeneous fluid mixture

a Numerical Study of Two-Dimensional Natural Convection in Square Open Cavities

Abstract: A numerical study of natural convection in a two-dimensional square open cavity under laminar steady-state conditions has been performed. The square cavity has one heated vertical wall facing a vertical opening and two insulated horizontal walls. Results are obtained for Rayleigh numbers ranging from 103 to 109 at unit Prandtl number with constant properties and the Boussinesq approximation. Heat transfer results approach those for an isothermal vertical flat plate at high Rayleigh numbers. Streamline and isotherm plots illustrate the effect of the open boundary on the basic flow patterns. A recirculation zone is found at high Rayleigh numbers above the bottom wall due to incoming flow turning around the corner. Thermally stratified flow below the top wall exits to form a wall plume.

The destruction of geochemical heterogeneities by differential fluid motions during mantle convection

Abstract: Summary. A two-dimensional numerical method is presented which permits the observation of deformation within convecting fluids at high Prandtl number. This method is applied to steady state convection, from which it is concluded that deformation is achieved by simple shear, when considered over long time-scales. In numerical examples over a wide range of Rayleigh numbers and heat sources this shear is shown usually to be prograde, with the periphery of the convection cell rotating slower than the core. Occasionally retrograde shear occurs, in which the reverse is true. When the method is applied to time-dependent convection, the deformation of discrete bodies occurs by lateral eddy diffusion of mass, and exponential increase of surface area. The rates of these two processes are coupled and vary as Ra0.5 over a wide range of Rayleigh numbers. It is shown that an eddy diffusive approximation is only appropriate for horizontal scales greater than about 3 times the depth of a convecting layer and for time-scales of at least 300 Myr in the upper mantle. The effects of variable viscosity and three-dimensionality are discussed, and the results of the experiments applied to the Earth's mantle. It is concluded that any convecting layer within the mantle must be well mixed on a lateral scale of at least 2000 km after a time of 0.5–1 Gyr, depending on the processes of magma extraction from a streaky parental material. The deformation rate depends only upon the layer depth and the fluid viscosity. Hence whole mantle convection is 5 times more effective in dispersing geochemical heterogeneities than is convection confined to the upper mantle. Various models for the spatial distribution of geochemical reservoirs are discussed and the only viable models are those in which geochemical reservoirs are identified with separate fluid layers. Whole mantle convection is not favoured by these conclusions. On other grounds, some layered models are discarded and we support a conventional model with separate convection in the upper and lower mantle with a boundary at about 700 km depth. Isotopic anomalies erupted at the surface must be relatively recent additions to the otherwise homogeneous upper mantle.

Variable Property Effects in Laminar Natural Convection in a Square Enclosure

Abstract: A numerical, finite-difference study has been carried out to determine the effects of variable properties on the temperature and velocity fields and the heat transfer rate in a differentially heated, two-dimensional square enclosure. Calculations have been done for a Rayleigh number range up to 106 , and temperature difference ratios θ0 = (TH –TC ) / TC of 0.2, 0.5, 1.0, and 2.0, where TH and TC are the hot and cold wall temperatures, respectively. Specific issues that have been addressed are the limits of validity of the Boussinesq approximation, the proper use of a reference temperature, variable property correlation of the heat transfer rate, and the limits of conduction-dominated regions.

Natural Convection in Open-Ended Inclined Channels

Abstract: Experiences de visualisation de l'ecoulement et du transfert de chaleur dans l'eau pour etudier l'effet de l'inclinaison θ sur la convection naturelle dans un canal a parois paralleles. On fait varier l'ecartement adimensionnel entre les parois (S/H), le mode de chauffage, et la difference de temperature entre la paroi et l'air ambiant (nombre de Rayleigh Ra s ). Obtention d'une correlation entre Nu s , Ra s , S/H et θ

Solutions and stability criteria of natural convective flow in an inclined porous layer

Abstract: Previous experiments on natural convection in a differentially heated porous layer with large lateral dimensions gave evidence for different configurations of flow. Depending on the values of the Rayleigh number, the inclination and the longitudinal extension of the layer, the three main structures observed correspond to a two-dimensional unicellular flow, polyhedral convective cells and longitudinal coils. In this paper there is a definition of the conditions necessary for these types of flow to exist using a linear stability theory and it is shown that the experimentally observed structures can be theoretically predicted by a three-dimensional numerical model based upon Galerkin's spectral method. Finally, the results of this model are used to show the influence of initial conditions on the setting up of the stationary flow.

Square-pattern convection in fluids with strongly temperature-dependent viscosity

Abstract: Three-dimensional numerical solutions are obtained describing convection with a square lattice in a layer heated from below with no-slip top and bottom boundaries. The limit of infinite Prandtl number and a linear dependence of the viscosity on temperature are assumed. The stability of the three-dimensional solutions with respect to disturbances fitting the square lattice is analysed. It is shown that convection in the form of two-dimensional rolls is stable for low variations of viscosity, while square-pattern convection becomes stable when the viscosity contrast between upper and lower parts of the fluid layer is sufficiently strong. The theoretical results are in qualitative agreement with experimental observations.

Finite element calculations of very high Rayleigh number thermal convection

Abstract: Summary. A finite element method with uniform and variable resolution meshes is used to model very high Rayleigh number Ra thermal convection in a square box of infinite Prandtl number, Boussinesq fluid with constant viscosity and thermodynamic properties. Heating is either entirely from below or mostly from within and the boundaries are stress free. The variable mesh is coarse in the interior of the convection cell and it is fine in the very thin boundary layers and plumes surrounding the core. The highest resolution variable mesh has a dimensionless grid spacing of 0.027 in the core and 0.0017 in the boundary layers. The boundary layers contain about 10 mesh points even at the highest values of Ra considered and are thus highly resolved. The variable mesh approach is shown to yield reliable simulations of convection as long as the aspect ratio of the most elongated boundary layer elements is not too large; values of about 4 to 6 work well. This aspect ratio also measures the increase in resolution in the boundary layers as compared with the central core. Steady single-cell rolls are computed for bottom heating and Ra up to 5 × 105 times the marginal instability value of the Rayleigh number Racr. One and two-cell roll solutions are calculated for f= 1, 0.8 and 0.6, where f is the fraction of the heat escaping through the top of the box that is generated internally. The values of Racr for f= 1, 0.8 and 0.6 are 1296, 1024 and 864, respectively. The largest of Ra/Racr at which unicellular convection is stable (steady) are approximately 390, 610 and 970, for f= 1, 0.8 and 0.6.

The departure from Darcy flow in natural convection in a vertical porous layer

Abstract: An analytical and numerical study is reported of steady‐state natural convection in a two‐dimensional porous layer heated from the side. Contrary to previous investigations of the phenomenon, which were all based on the Darcy flow model, a vector generalization of Forchheimer’s one‐dimensional model is used in the present study, which is valid for all values of local Reynolds number based on pore size. A matched boundary layer solution of the type developed by Weber for Darcy flow is developed for the limit of large‐pore Reynolds numbers (the ‘‘non‐Darcy’’ limit). It is shown that the natural convection phenomenon in the non‐Darcy limit is governed by a new dimensionless group, the Rayleigh number for the higher Reynolds number limit, Ra∞. Numerical experiments are reported in the range 1.6×105≤Ra∞≤1.6×109, in a porous layer with height/thickness ratio equal to 2, and with a high value of Darcy modified Rayleigh number (Ra=4000). The numerical experiments confirm the flow features and scales anticipated by the matched boundary layer solution for the non‐Darcy limit. The experiments also document the transition from the well‐known Darcy flow to the large‐pore Reynolds‐number limit treated in this paper.

Natural convection flow of a non-Newtonian fluid between two vertical flat plates

Abstract: The natural convection of a homogeneous incompressible fluid of grade three is investigated between two infinite parallel vertical plates. The effect of the non-Newtonian nature of fluid on the skin friction and heat transfer are studied.

Experimental investigation of combined forced and free convection in horizontal and inclined tubes

Abstract: This investigation considers the effects of free convection on the laminar flow of water through a circular duct having essentially constant wall heat transfer rate per unit length of the duct and circumferentially uniform wall temperature. The effect of the Reynolds and Rayleigh number variations on heat transfer results has been analysed for both horizontal and inclined pipe. The experiment has covered the range of the inlet Reynolds number from 200 to 2300, and of the Rayleigh number from 6,000 to 70,000. The effect of pipe inclination has been investigated for slope angle values up to 60°, with laminar ascending flow.

Thermal convection under external modulation of the driving force. I. The Lorenz model

Abstract: Thermal convection between horizontal plates is considered for a situation in which the driving force varies periodically in time. This variation may come from changes in the temperature or the plates or from a vertical oscillation of the cell, causing variation of the gravitational force. Truncation of the Boussinesq equations leads to a three-mode model which is a generalization of the Lorenz model to the case of external modulation. Similar models have previously been introduced by Finucane and Kelly for stress-free horizontal boundaries and by Gresho and Sani for the rigid-boundary case. The threshold behavior is that of a parametrically driven damped oscillator, whose bifurcations are studied numerically, as well as analytically in certain limits. It is found that in general the modulation stabilizes the conducting state. For stress-free horizontal boundaries the threshold shifts predicted by the model coincide with the results of Rosenblat and Herbert in the limit of low frequency and agree well with Venezian's results for small modulation amplitude, both obtained using the full Boussinesq equations. For rigid boundaries the results agree well with numerical calculations of Rosenblat and Tanaka. The nonlinear behavior of the model is also studied, and the convective contribution to the heat current evaluated. The Lorenz model is shown to reproduce, either exactly or to a good approximation, most previous theoretical results on modulated convection, and the model can be studied simply for a wide range of parameters. The above discussion refers to a laterally infinite system. For a real finite system, sidewall effects are shown to cause a rounding of the convective threshold in the presence of modulation, particularly at low frequencies. A calculation of these effects is carried out within the framework of the Lorenz truncation, and the resulting imperfect bifurcation of the model is studied numerically. In a companion paper (immediately following this one) quantitative experimental results are presented and compared to the predictions of the Lorenz model.

Square versus roll pattern at convective threshold.

Abstract: We present observations and measurements of convection between two moderately conducting plates. A staionary square pattern is observed just at the convective threshold. But at high Rayleigh number the preferred convective pattern becomes the usual roll structure. The transition from squares to rolls involves an unexpected Hopf bifurcation.

Pattern Formation and Wave-Number Selection by Rayleigh-Bénard Convection in a Cylindrical Container

Abstract: We describe an apparatus and procedures for simultaneous heat transport measurements and computer enhanced shadowgraph flow-pattern imaging in a shallow horizontal layer of fluid heated from below. The heat transport measurements have a resolution of better than 0.1%, and the shadowgraph technique can detect the flow field for ≡ (R – Rc)/Rc as small as 10-2 (Rc is the critical value of the Rayleigh number R for onset of convection). The apparatus and procedures were used to study pattern and wave-number evolution in a cylindrical layer of water with radius-to-height ratio L = 7.5 and Prandtl number σ = 6.1. We found that dynamic sidewall forcing during the early thermal transients after a change in the heat current from a subcritical to a supercritical value establishes a cylindrical flow pattern. Once created, this pattern is stable in our apparatus over the wide range 0.16 8 even after the transients have decayed. With changing , adjustment in the wave number k takes place discontinuously by hysteretic changes at the cell center in the number of convection roll pairs. When is increased, the discontinuous changes at the cell center are towards smaller k and are preceded by a continuous loss of cylindrical symmetry (the middle roll pair moves off center). The selected wave numbers coincide neither with the zig-zag instability of the infinite system, as once suggested, nor with a linear extrapolation to = 0(1) of the recent prediction to lowest order in of Manneville and Piquemal and of Cross. Comparison of the selected k with measurements by others reveals no dependence upon L and σ. For < 0.16, the cylindrical pattern is unstable and decays on a time scale much longer than a horizontal diffusion time to patterns of rolls which tend to be perpendicular to the sidewalls and which contain defects. Once formed, these latter patterns will persist at large values of . These patterns also undergo a wave-number adjustment process with hysteretic changes mediated mostly by focus singularities near the walls. In these cases, larger values of also tend to produce smaller values of k.