Showing papers on "Rayleigh number published in 1987"

Natural Convection in a Vertical Porous Cavity: A Numerical Study for Brinkman-Extended Darcy Formulation

Abstract: A dimensional analysis of the Brinkman-extended Darcy formulation, which includes the transport and viscous terms, leads to four governing parameters for steady-state natural convection in a vertical porous cavity. They are: Rayleigh number, Darcy number, diffusion parameter {Omega}, and aspect ratio. Numerical results for 0 {le} Da {le} 10{sup {minus}1}, 10 {le} Ra* {le} 5 {times} 10{sup 3}, and A = 1 and 5, indicate that the temperature and velocity fields are significantly modified, the flow regimes are delayed, and the heat transfer rate is decreased when the Darcy number is increased beyond 10{sup {minus}5} for fixed Ra{sup {asterisk}} and A. The slope of the In (Nu) versus In (Ra*) curve in the boundary layer regime decreases from 0.53 at Da = 0 to 0.264 at Da = 10{sup {minus}1} when A = 5. The contribution of the transport term increases with {Omega}, Da, and Ra*, but the effect on the overall heat transfer is insignificant. However, the problem becomes ill formulated at high values of these parameters and may require the consideration of Forchheimer modifications. A scale analysis is also presented to show that the inertia term is of a low order of magnitude in comparison with the viscousmore » term at high Prandtl numbers.« less

On the onset of convection in rotating spherical shells

Abstract: The linear problem of the onset of convection in rotating spherical shells is analysed numerically in dependence on the Prandtl number. The radius ratio η=r i/r o of the inner and outer radii is generally assumed to be 0.4. But other values of η are also considered. The goal of the analysis has been the clarification of the transition between modes drifting in the retrograde azimuthal direction in the low Taylor number regime and modes traveling in the prograde direction at high Taylor numbers. It is shown that for a given value m of the azimuthal wavenumber a single mode describes the onset of convection of fluids of moderate or high Prandtl number. At low Prandtl numbers, however, three different modes for a given m may describe the onset of convection in dependence on the Taylor number. The characteristic properties of the modes are described and the singularities leading to the separation with decreasing Prandtl number are elucidated. Related results for the problem of finite amplitude convec...

Compositional and thermal convection in magma chambers

Abstract: Magma chambers cool and crystallize at a rate determined by the heat flux from the chamber. The heat is lost predominantly through the roof, whereas crystallization takes place mainly at the floor. Both processes provide destabilizing buoyancy fluxes which drive highly unsteady, chaotic convection in the magma. Even at the lowest cooling rates the thermal Rayleigh number Ra is found to be extremely large for both mafic and granitic magmas. Since the compositional and thermal buoyancy fluxes are directly related it can be shown that the compositional Rayleigh number Rs (and therefore a total Rayleigh number) is very much greater than Ra. In the case of basaltic melt crystallizing olivine Rs is up to 106 times greater than Ra. However compositional and thermal buoyancy fluxes are roughly equal. Therefore thermal and compositional density gradients contribute equally to convection velocities in the interior of the magma. Effects of thermal buoyancy generated by latent heat release at the floor are included. The latent heat boundary layer at the floor of a basaltic chamber is shown to be of the order of 1 m thick with very low thermal gradients whereas the compositional boundary layer is about 1 cm thick with large compositional gradients. As a consequence, the variation in the degree of supercooling in front of the crystal-liquid interface is dominated by compositional effects. The habit and composition of the growing crystals is also controlled by the nature of the compositional boundary layer. Elongate crystals are predicted to form when the thickness of the compositional boundary layer is small compared with the crystal size (as in laboratory experiments with aqueous solutions). In contrast, equant crystals form when the boundary layer is thicker than the crystals (as in most magma chambers). Instability of the boundary layer in the latter case gives rise to zoning within crystals. Diffusion of compatible trace elements through the boundary layer can also explain an inverse correlation, observed in layered intrusions, between Ni concentration in olivine and the proportion of Ni-bearing phases in the crystallizing assemblage.

Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vacillating flow

Abstract: The instabilities of convection columns (also called thermal Rossby waves) in a cylindrical annulus rotating about its axis and heated from the outside are investigated as a function of the Prandtl number P and the Coriolis parameter η* When this latter parameter is sufficiently large, it is found that the primary solution observed at the onset of convection becomes unstable when the Rayleigh number exceeds its critical value by a relatively small amount Transitions occur to columnar convection which is non-symmetric with respect to the mid-plane of the small-gap annular layer Further transitions introduce convection flows that vacillate in time or tend to split the row of columns into an inner and an outer row of separately propagating waves Of special interest is the regime of non-symmetric convection, which exhibits decreasing Nusselt number with increasing Rayleigh number, and the indication of a period doubling sequence associated with vacillating convection

Magnetic and Gravitational Natural Convection of Melted Silicon—Two-Dimensional Numerical Computations for the Rate of Heat Transfer : Heat Transfer, Combustion, Power, Thermophysical Properties

Abstract: The two-dimensional natural convection of fluid under both a magnetic and a gravitational field was modeled by conservation equations. Sample computations were carried out for the fluid in a square enclosure for Rayleigh number of from 104 to 106, for Hartman number of from 1 to 103 and for Prandtl number equal to 0.054, equivalent to melted silicon. The numerical computations converged successfully and the Nusselt numbers obtained were correlated to give an empirical equation for the rate of heat transfer. The steady state solutions were graphically visualized. At Ha=103 and Ra=106, the point at which the temperature profile was almost linear, the flow was almost suppressed and elongated in a regime with high wave numbers.

Nonlinear transition in three-dimensional convection

Abstract: Steady and oscillatory convection in a rectangular box heated from below are studied by means of a numerical solution of the three-dimensional, time-dependent Boussinesq equations. The effect of the rigid sidewalls of the box on the spatial structure and the dynamical behaviour of the flow is analysed. Both conducting and adiabatic sidewalls are considered. Calculated streamlines illustrate the three-dimensional structure of the steady flow with Prandtl numbers 0.71 and 7. The onset and the frequency of the oscillatory instability are calculated and compared with available experimental and theoretical data. With increasing Rayleigh number a subharmonic bifurcation and the onset of a quasi-periodic flow can be observed. A comparison of the different time-dependent solutions shows some interesting relations between the spatial structure and the dynamical behaviour of the confined flow.

A General Similarity Transformation for Combined Free and Forced-Convection Flows Within a Fluid-Saturated Porous Medium

Abstract: It is the purpose of the present paper to introduce a general transformation procedure appropriate to the problem of combined free and forced convection in a porous medium. It will be shown that particular transformations proposed in the previous papers by Cheng and Minkowycz and co-workers are simply the specific forms of the present general transformation. Pure forced convection will be treated first as a limiting case of combined free and forced convection. The analysis reveals that any two-dimensional or axisymmetric body of arbitrary shape possesses its corresponding class of wall temperature distributions which permit similarity solutions. Secondly, combined free and forced convection will be considered to seek similarity solutions. It is found that, unlike in pure forced convection, similarity solutions in mixed convection are possible only when the external free-stream velocity varies every where in proportion to the product of the streamwise component of the gravity force and the wall-ambient temperature difference.

Some geodynamical effects of anisotropic viscosity

Abstract: Summary. Rheological anisotropy in the mantle may arise from various causes, the most important are the preferred orientation of single crystals, and the orientation of streaks of eclogitic material in a ‘marble cake’ mantle. In both cases the orientation can be achieved by the flow kinematics, and in steady state convection the planes of easy slip will eventually be orientated parallel to the stream lines. The dynamic equations for a 2-D, incompressible, anisotropic viscous fluid are derived, and some geodynamic consequences are studied. A difference of the order 10-100 between ‘shear’ and ‘normal’ viscosity is considered. Post-glacial rebound would be affected by the mechanical properties in deeper parts of the mantle. The observed uplift history near the ice margin has been taken as argument against strong viscosity stratification in the mantle. It is shown here that with strong viscous anisotropy at least in the upper mantle, the uplift history is compatible with a significant increase in viscosity between upper and lower mantle. The topographic and geoid signal can show a spatial offset from a generating mass anomaly in the mantle. This may provide the opportunity for in situ observation of anisotropic flow properties in the mantle. In numerical case studies of steady state convection streamline-orientated anisotropy is considered. At high Rayleigh number for bottom-heated convection the formation of velocity boundary layers is observed, with no flow in the stagnant core of the convection cell. With internal heating or strong pressure- and temperature-dependence of viscosity this is no longer found, but in the latter case anisotropy enhances the channelling of flow into lowviscosity regions, e.g. hot rising plumes. A few experiments have been performed for time-dependent convection. The results suggest that the onset of time-dependent boundary layer instabilities may be somewhat retarded, but would not be strongly suppressed by anisotropy. Strong time-dependence may destroy any large-scale coherence in the orientation of anisotropy and thus render the mantle to appear viscously isotropic.

Natural convection about a heated horizontal cylinder in a porous medium

Abstract: The natural convection from a heated circular cylinder in an unbounded region of porous medium is investigated for the full range of Rayleigh numbers. At small Rayleigh numbers a qualitative solution is obtained and at large Rayleigh numbers the second-order boundary-layer solution is found that takes into account the first-order plume solution. In order to find the solution at finite Rayleigh numbers the two governing coupled, nonlinear, elliptic partial differential equations are expressed in finite-difference form using a specialized technique which is second-order accurate everywhere. Further, methods are devised which deal with the plume and infinity boundary conditions. Although numerical results are presented for Rayleigh numbers up to 400 solutions of the finite-difference equations can be obtained for higher values of the Rayleigh numbers but in these cases the mesh size used is too large to adequately deal with the developing boundary-layer on the cylinder and the plume.The numerical results show how the theories at both low and high Rayleigh numbers are approached. The plume solution which develops with increasing Rayleigh number agrees with that predicted by the theory presented using the boundary-layer approximation. No separation of the flow at the top of the cylinder is found and there are no indications that it will appear at higher values of the Rayleigh number. The results presented here give reasonable agreement with the existing experimental results for Rayleigh numbers of order unity. However as the Rayleigh number increases to order 102 there is a large discrepancy between the theoretical and experimental results and this is because at these higher values of the Rayleigh number the Darcy approximation has been violated in the experimental results. This indicates the severe limitations of some of the existing theories which use boundary-layer analyses and the Darcy approximation for flows in a porous medium. The application of Darcy's law requires that the size of the pores be much smaller than the scale of the bulk flow and inertial and thermal lengthscales.

Throughflow effects in the Rayleigh-Bénard convective instability problem

Abstract: The effect of vertical throughflow on the onset of convection in a fluid layer, between permeable horizontal boundaries, when heated uniformly from below, is re-examined analytically. It is shown that when the Peclet number Q is large in magnitude, the critical Rayleigh number Rc is proportional to Qn, where n = 0, 1, 2, 3 or 4, with a coefficient depending on the Prandtl number P, according to the types of boundaries. When the upper and lower boundaries are of different types, the effect of a small amount of throughflow in one direction is to decrease Rc. This is so when the throughflow is away from the more restrictive boundary. Contributions arise from the curvature of the basic temperature profile, and from the vertical transport of perturbation velocity and perturbation temperature. The decrease in Rc is small if P ∼ 1 but can be of significant size if P [Lt ] 1 or P [Gt ] 1.

Vorticity-Velocity Method for the Graetz Problem and the Effect of Natural Convection in a Horizontal Rectangular Channel With Uniform Wall Heat Flux

Abstract: Numerical solutions given by a vorticity-velocity method are presented for combined free and forced laminar convection in the thermal entrance region of a horizontal rectangular channel without the assumptions of large Prandtl number and small Grashof number. The channel wall is heated with a uniform wall heat flux. Typical developments of temperature profile, secondary flow, and axial velocity at various axial positions in the entrance region are presented. Local friction factor and Nusselt number variations are shown for Rayleigh numbers Ra = 10{sup 4}, 3 {times} 10{sup 4}, 6 {times} 10{sup 4}, and 10{sup 5} with the Prandtl number as a parameter. The solution for the limiting case of large Prandtl number and small Grashof number obtained from the present study confirms the data of existing literature. It is observed that the large Prandtl number assumption is valid for Pr = 10 when Ra {le} 3 {times} 10{sup 4} but for a larger Prandtl number when the Rayleigh number is higher.

Three-dimensional axisymmetric model for convection in laser-melted pools

Abstract: A three-dimensional axisymmetric model of the fluid flow and heat transfer in a laser-melted pool is developed. The model corresponds to the limiting case when the scanning velocity is small compared with the recirculating velocity. This model is also valid for spot welding. Non-dimensional forms of the governing equations are derived, from which four dimensionless parameters are obtained: the Marangoni number, the Prandtl number, the dimensionless melting temperature, and the radiation factor. Their effects and significance are discussed, and numerical solutions are obtained. The position and shape of the solid/liquid interface are obtained by an iterative scheme. The quantitative effects of the dimensionless parameters on pool shape are presented. In the presence of the flow field, the heat transfer becomes convection dominated. The effect of convection on isotherms within the molten pool is discussed, and experimental results are presented.MST/535

Nonlinear oscillatory convection

Abstract: A numerical analysis has been performed of three-dimensional time-dependent solutions which bifurcate supercritically from two-dimensional convection-roll solutions at the onset of the oscillatory instability. The bifurcating solutions describe a periodic shifting forward and backward of the convection rolls and lead to a strong deformation of the rolls as the Rayleigh number increases. Since the bifurcating solution is stable in the form of a travelling wave, the computational expense can be reduced by assuming a moving coordinate. Travelling-wave solutions have been computed in the case of rigid boundaries as a function of the Prandtl number and of the two basic wavenumbers αx, αy of the problem. The onset of oscillations reduces the heat transport in comparison with that of two-dimensional rolls because the occupation of a new degree of freedom of motion by the oscillation reduces the energy of the heat-transporting component of convection. A limited stability analysis of finite-amplitude travelling waves has been performed and the onset of an asymmetric mode of oscillations is determined as a function of the parameters of the problem. This mode appears to be identical with a mode that was observed in the numerical simulations of Lipps (1976) and McLaughlin & Orszag (1982).

Three-dimensional numerical simulation of convection in low-Prandtl-number fluids

Abstract: We present three-dimensional numerical simulations of convection in a low-Prandtlnumber fluid confined between two infinite horizontal bounding surfaces maintained at constant temperatures. We consider the case of free-slip boundary conditions for a fluid of Prandtl number Pr = 0.2 and that of rigid boundary conditions with Pr = 0.025. In the former situation, we observe stationary, periodic, biperiodic and chaotic regimes as the Rayleigh number is increased. In the later situation, the dynamics involves very different characteristic times, and only stationary and time-periodic solutions have been simulated. Convergence to the later regime may occur after a long transient where the amplitude of the oscillation is slowly modulated.

Time‐dependent convection in elongated Rayleigh‐Benard cells

Abstract: The onset of time-dependence and some subsequent transitions in two-dimensional Rayleigh-Benard cells of aspect ratios a=2.5 and a=4 are studied numerically for a Boussinesq fluid of infinite Prandtl number. Oscillatory convection starts at a Rayleigh number of only 21,000 (a=2.5) or 12,000 (a=4.0) for stress-free boundaries. Above Ra=100,000 the cells with a=4.0 break down into several short cells, whereas single cells with a=2.5 remain stable to at least Ra=500,000. With rigid boundaries the tendency towards break-up into short cells is found to be more pronounced. Long-wavelength cells, modulated by travelling boundary-layer instabilities, appear to be a likely form of convection in the earth's mantle.

Prandtl Number Dependence of Natural Convection in Porous Media

Abstract: The effect of Prandtl number of a medium on heat transfer across a horizonal layer was measured. Stainless steel particles of diameters 1.6, 3.2, and 4.8 mm, glass particles of diameters 2.5 and 6.00 mm, and lead particles of diameter 0.95 mm were used with silicon oil, water and mercury as working fluids. The bed height was varied from 2.5 to 12 cm. Experimental results are presented showing Nusselt number as a function of medium Rayleigh number with the effective Prandtl number, defined as the product of medium Prandtl number and Kozeny--Carmen constant, serving as a parameter. Correlations for Nusselt number are given for effective Prandtl number less than 0.1 and for effective Prandtl number greater than 0.1, which corresponds to an infinite effective Prandtl number. For the steel--water case the wavenumber is shown as a function of medium Rayleigh number.

Hopf Bifurcation in the Double-Glazing Problem With Conducting Boundaries

Abstract: Oscillatory convection has been observed in recent experiments in a square, air-filled cavity with differentially heated sidewalls and conducting horizontal surfaces. The author shows that the onset of the oscillatory convection occurs at a Hopf bifurcation in the steady-state equations for free convection in the Boussinesq approximation. The location of the bifurcation point is found by solving an extended system of steady-state equations. The predicted critical Rayleigh number and frequency at the onset of oscillations are in excellent agreement with the values measured recently and with those of a time-dependent simulation. Four other Hopf bifurcation points are found near the critical point and their presence supports a conjectured resonance between traveling waves in the boundary layers and interior gravity waves in the stratified core.

Order and fluctuations in nonequilibrium molecular dynamics simulations of two-dimensional fluids

Abstract: Finite systems of hard disks placed in a temperature gradient and in an external constant field have been studied, simulating a fluid heated from below. We used the methods of nonequilibrium molecular dynamics. The goal was to observe the onset of convection in the fluid. Systems of more than 5000 particles have been considered and the choice of parameters has been made in order to have a Rayleigh number larger than the critical one calculated from the hydrodynamic equations. The appearance of rolls and the large fluctuations in the velocity field are the main features of these simulations.

Natural Convection of Non-Newtonian Fluids About a Horizontal Surface in a Porous Medium

Abstract: The problem of natural convection of a non-Newtonian power-law fluid about a horizontal impermeable surface in the porous medium is considered, where the plate is assumed with a nonuniform heat flux distribution. The present study is based on the boundary layer approximation and only suitable for a high Rayleigh number. Similarity solutions are obtained by using the fourth-order Runge-Kutta method and the Nachtsheim-Swigert iteration scheme. The effects of the nonuniform wall heat flux q w (x) and the new power-law index n on the heat transfer characteristics are discussed.

A mechanism for differential rotation based on angular momentum transport by compressible convection

Abstract: It is shown that compressible convective motions in a density-stratified rotating fluid give rise to non-diffusive fluxes of the angular momentum (Λ-effect), thus leading to differential rotation. This also occurs for the case of extremely small convection anisotropy. Attention is focused on this case of quasi-isotropic convection and the Λ-effect of quasi-isotropic convection is derived. The corresponding fluxes of the angular momentum, as well as the resulting deviation of the rotation from homogeneity, are shown to be proportional to the second spatial derivative of the fluid density squared. The solution of the Reynolds equation is found for a particular density profile. Isorotational surfaces corresponding to different values of the Rossby number, Ro, are considered. The angular velocity increases with depth and towards the equator when Ro≾0.3 which corresponds to the middle and lower parts of the solar convection zone. The results obtained are compared with observations. A possible physical...

Viscous and Inertial Convection at Low Prandtl Number: Experimental Study

Abstract: Rayleigh-Benard convection in mercury is studied in the vicinity of its threshold with a high-resolution experimental system. The Prandtl number of the fluid is varied between 0.02 and 0.04. The experimental evidence of two different stationary convective regimes, called viscous and inertial, is reported. Their dependence on the Prandtl number P is compared with the theoretical predictions in the small Prandtl-number limit.