Showing papers on "Rayleigh number published in 1984"

Convection Heat Transfer

Abstract: Fundamental Principles Laminar Boundary Layer Flow Laminar Duct Flow External Natural Convection Internal Natural Convection Transition to Turbulence Turbulent Boundary Layer Flow Turbulent Duct Flow Free Turbulent Flows Convection with Change of Phase Mass Transfer Convection in Porous Media.

Convection with pressure- and temperature-dependent non-Newtonian rheology

Abstract: Summary. The influence of non-linear stress-strain rate relation on thermal convection in a fluid whose rheological properties are also pressure- and temperature-dependent is studied in a series of numerical models. A finite element method with an upwind weighted residual technique and B splines is applied to obtain steady state and time-dependent solutions of two-dimensional convection in rectangular enclosures. A cubic power law rheology is taken in most cases and the result is compared to Newtonian convection. The model calculations comprise different effective Rayleigh numbers in the range of 104-106, different values for the activation energy and volume which determine the p, T dependence, different modes of heating, different aspect ratios, and different mechanical boundary conditions. In almost all cases power-law rheology leads to considerably different flow patterns and heat transfer properties than those predicted for Newtonian convection. In general the effect is to reduce the internal viscosity contrast which is produced by the pressure and temperature effect. This can in some cases mobilize regions of the cell which are stagnant in the case of Newtonian rheology. The properties of stationary power-law convection can be imitated closely by Newtonian flow with a reduced value for the activation enthalpy βH* with β= 0.3-0.5. If the stress exponent is greater than 3, the p, T influence is reduced even more. In the time-dependent regime non-linear rheology enhances oscillatory behaviour, increasing the amplitude and leading to peak-line maxima and broad minima. The question of proper evaluation of the average viscosity or effective Rayleigh number in variable viscosity convection is also addressed. Empirically it was found to be more satisfactory to weight the local viscosity with the absolute value of the strain rate rather than with its square as was proposed earlier. Further points under consideration were the effects of combined linear and non-linear rheology and the parametrization of variable viscosity convection. It appears possible that non-Newtonian convection plays a key role in determining the convective style in a planetary mantle. Non-Newtonian convection might be essential for facilitating flow in the Earth's lower mantle, for making the upper thermal boundary layer behave like a quasi-rigid but subductable plate, or for stabilizing a hot low-viscosity boundary layer.

Rayleigh-bénard convection

Abstract: This paper presents a physicist's approach to Rayleigh-Benard convection widely illustrated with experimental results. The basis of the mechanism of the instability is simply presented with physical reasons for the existence of a critical threshold. A detailed examination of the spatial organization is made with discussion of ordered and disordered structures. Furthermore it is shown that the measurements of the local velocity give a good quantitative description of the convective state. A complete parallel between the Rayleigh-Benard convection near onset and a critical phenomenon is given in the framework of a mean field approach, including both spatial as well as temporal effects. Non-Boussinesq convection is presented as symmetry breaking, changing the second order transition into a (partially) first order transition. The last section is devoted to the ever present but still not completely understood question of the dynamics of the convective pattern; the importance of the existence and motio...

Observed Flow Reversals and Measured-Predicted Nusselt Numbers for Natural Convection in a One-Sided Heated Vertical Channel

Abstract: A three-part study encompassing both experiment and analysis has been performed for natural convection in an open-ended vertical channel. One of the principal walls of the channel—the heated wall—was maintained at a uniform temperature, while the other principal wall was unheated. The experiments, which included flow visualization and Nusselt number measurements, were carried out with water in the channel and in the ambient which surrounds the channel. At Rayleigh numbers which exceeded a threshold value, the visualization revealed a pocket of recirculating flow situated adjacent to the unheated wall in the upper part of the channel. The recirculation was fed by fluid drawn into the top of the channel, adjacent to the unheated wall. Average Nusselt numbers for the heated wall were measured over a three orders of magnitude range of a single correlating parameter, which includes the Rayleigh number and the ratio of the channel length to the interwall spacing. The Nusselt numbers were found to be unaffected by the presence of the recirculation zone. Numerical solutions obtained via a parabolic finite difference scheme yielded Nusselt numbers in good agreement with those of experiment. The numerical results covered the Prandtl number range from 0.7 to 10.

Experiments on transient natural convection in a cavity

Abstract: A laboratory experiment is used to study the transient flow in an initially isothermal cavity at temperature T0 following the rapid change of the two vertical endwalls to temperatures T0 ± ΔT respectively. Individual temperature records are taken and the transient flow in the entire cavity is visualized with the aid of a tracer technique. It is shown that an oscillatory approach to final steady-state conditions exists for certain flow regimes, although the form of the oscillatory response is different to that suggested by previous work. It is argued that this oscillatory behaviour is due to the inertia of the flow entering the interior of the cavity from the sidewall boundary layers, which may lead to a form of internal hydraulic jump if the Rayleigh number is sufficiently large.

Bénard convection with time‐periodic heating

Abstract: A thin liquid layer, which is heated from below, has its lower boundary modulated sinusoidally in time with amplitude δ. Weakly‐nonlinear stability theory shows that the modulation produces a range of stable hexagons near the critical Rayleigh number. For small δ the range is O(δ4) in size and decreases with modulation frequency. These hexagons bifurcate subcritically and correspond to downflow at cell centers.

Pattern selection in single-component systems coupling Bénard convection and solidification

Abstract: A horizontal layer is heated from below and cooled from above so that the enclosed single-component liquid is frozen in the upper part of the layer. When the imposed temperature difference is such that the Rayleigh number across the liquid is supercritical, there is Benard convection coupled with the dynamics of the solidification interface. An experiment is presented which shows that the interfacial corrugations that result are two-dimensional when this solid is thin but hexagonal when the solid is thick. A weakly nonlinear convective instability theory is presented which explains this behaviour, and isolates this ‘purely thermal’ mechanism of pattern selection. Jump behaviour is seen in the liquid-layer thickness at the onset of hexagonal convection.

A nonlinear stability analysis of the Bénard–Marangoni problem

Abstract: A nonlinear analysis of Benard–Marangoni convection in a horizontal fluid layer of infinite extent is proposed. The nonlinear equations describing the fields of temperature and velocity are solved by using the Gorkov–Malkus–Veronis technique, which consists of developing the steady solution in terms of a small parameter measuring the deviation from the marginal state. This work generalizes an earlier paper by Schluter, Lortz & Busse wherein only buoyancy-driven instabilities were handled. In the present work both buoyancy and temperature-dependent surface-tension effects are considered. The band of allowed steady states of convection near the onset of convection is determined as a function of the Marangoni number and the wavenumber. The influence of various dimensionless quantities like Rayleigh, Prandtl and Biot numbers is examined. Supercritical as well as subcritical zones of instability are displayed. It is found that hexagons are allowable flow patterns.

Instabilities of convection rolls with stress-free boundaries near threshold

Abstract: The stability properties of steady two-dimensional solutions describing convection in a horizontal fluid layer heated from below with stress-free boundaries are investigated in the neighbourhood of the critical Rayleigh number. The region of stable convection rolls as a function of the wavenumber α and the Rayleigh number R is bounded towards higher α by the monotonic skewed varicose instability, while towards low wavenumbers stability is limited by the zigzag instability or by the oscillatory skewed varicose instability. Only for a limited range of Prandtl numbers, 0·543 < P < ∞, does a finite domain of stability exist. In particular, convection rolls with the critical wavenumber αc are always unstable.

Thermosolutal Convection during Directional Solidification

Abstract: During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

Experimental Investigation of Natural Convection Losses From Open Cavities

Abstract: Experimental results for natural convection in a cavity are reported. Both constrained andd unconstrained cavity geometries were studied. Detailed velocity profiles were obtained using Laser doppler velocimetry for Rayleigh numbers between 3 x 10/sup 10/ and 2 x 10/sup 11/, corresponding to a constant elevated wall temperature boundary condition. Characteristics of two-dimensional and three-dimensional flows obtained with dye flow visualization are discussed, including boundary layer transition to turbulence, flow patterns in the cavity, and flow outside of the cavity. Local Nusselt number is correlated with local Rayleigh number for constrained and unconstrained cavities.

The transition from roll to square-cell solutions in Rayleigh–Bénard convection

Abstract: We consider three-dimensional finite-amplitude thermal convection in a fluid layer with boundaries of finite conductivity. Busse & Riahi (1980) and Proctor (1981) showed that the preferred planform of convection in such a system is a square-cell tesselation provided that the boundaries are much poorer conductors than the fluid, in contrast to the roll solutions which are obtained for perfectly conducting boundaries. We determine here the conductivity of the boundaries at which the preferred planform changes from roll to square-cell type. We show that, for low-Prandtl-number fluids (e.g. mercury), square-cell solutions are realized only when the boundaries are almost insulating; while, for high-Prandtl-number fluids (e.g. silicone oils), square-cell solutions are stable when the boundaries have conductivity comparable to that of the fluid.

An experimental approach to thermal convection in a two‐layered mantle

Abstract: If the 650 km discontinuity marks a compositional boundary, as has been suggested, then the upper and lower mantle may be convecting separately. A series of laboratory experiments on two-layered convection were made in order to determine how thermal convection interacts with a stable density discontinuity. The working fluid consisted of two superposed layers of GLOBE 1132 syrup, a glucose solution with a Newtonian viscosity which depends strongly on temperature. The initial density contrast between layers ranged from 0.5% to 8%. A uniform heat flux was supplied to the base of the lower layer. By varying the heat flux, Rayleigh numbers between 4× 04 and 1×107 were obtained. In every case, two-layered convection was observed, but in no case did a steady state result. Instead, a slow mixing between the layers occurred, driven by viscous stresses acting on the density interface. The mixing mechanism was provided by convective eddies which entrained fluid across the discontinuity in the form of thin schlieren. Mixing continued until the density contrast across the discontinuity became small enough to permit overturning. The mixing rate was determined by monitoring changes in dye concentration in each layer. It is found that the mixing rate is governed by the bulk Richardson number Ri, a measure of the ratio of between interfacial buoyancy and viscous forces. Mixing rate data from experiments covering the range 80

Prandtl number effect on benard convection in porous media.

Abstract: A numerical study of buoyance-driven two-dimensional convection in a fluid-saturated horizontal porous layer is reported emphasizing the nonlinear inerital effect on heat transport. The Forchheimer-Brinkman-Darcy-Boussinesq formulation and a single energy equation for the volume-average temperature are used. Closure to the wavenumber selection problem is sought through a criterion based on the Glansdorff and Prigogine theory of nonequilibrium thermodynamics. Good agreement with laboratory data and the analogy with th Rayleigh-Benard problem are corroborative facts which justify smililar non-Darcian formulations and demonstrate the role of the quadratic inertial terms in decreasing the mean convective heat transfer across the layer.

The Stability of Natural Convection in a Vertical Layer of Dielectric Fluid in the Presence of a Horizontal ac Electric Field

Abstract: Linear stability theory is applied to examine the effect of a horizontal ac electric field on the stability of natural convection which occurs in a dielectric fluid between two parallel vertical plates maintained at different temperatures. The power series method is used to obtain the eigenvalue equation which is then solved numerically. It is shown that when the electrical Rayleigh number L (defined in the text) is less than about 2130 the electric field has no effect on the stability of natural convection, and that when L exceeds this value the electric field and the natural convection flow are coupled strongly and they exhibit a complicated but interesting effect on the stability of the system. A rough criterion is given for applicability of the power series method to the present problem.

Multiple buoyancy driven flows in a vertical cylinder heated from below

Abstract: The structure of axisymmetric buoyancy-driven convection in a vertical cylinder heated from below is probed by finite element solution of the Boussinesq equations coupled with computed-implemented perturbation techniques for detecting and tracking multiple flows and for determining flow stability. Results are reported for fluids with Prandtl number of one and for cylinders with aspect ratio (Lambda) (defined as the height to radius of the cylinder) between 0.5 and 2.25. Extensive calculations of the neutral stability curve for the static solution and of the nonlinear motions along the bifurcating flow families show a continuous evolution of the primary cellular motion from a single toroidal cell to two and three cells nested radially in the cylinder, instead of the sharp transitions found for a cylinder with shear-free sidewalls. The smooth transitions in flow structure with Rayleigh number and lambda are explained by nonlinear connectivity between the first two bifurcating flow families formed either by a secondary bifurcation point for Lambda or = Lambda * approximately 0.80 or by a limit point for Lambda Lambda *. The transition between these two modes may be described by the theory of multiple limit point bifurcation.