Showing papers on "Rayleigh number published in 1991"

Natural convection in a mushy layer

Abstract: Governing equations for a mushy layer are analysed in the asymptotic regime Rm [Gt ] 1, where Rm is an appropriately defined Rayleigh number. A model is proposed in which there is downward flow everywhere in the mushy layer except in and near localized chimneys, which are characterized by having zero solid fraction. Upward, convective flow within the chimneys is driven by compositional buoyancy. The radius of each chimney is determined locally by thermal balances within a boundary layer that surrounds it. Simple solutions are derived to determine the structure of the mushy layer away from the immediate vicinity of chimneys in order to demonstrate the gross effects of convection upon the solidification within the layer.

Numerical simulations of three-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity

Abstract: Numerical calculations are presented for the steady three-dimensional structure of thermal convection of a fluid with strongly temperature-dependent viscosity in a bottom-heated rectangular box. Viscosity is assumed to depend on temperature T as exp ( --ET), where E is a constant ; viscosity variations across the box r (= exp (E)) as large as lo5 are considered. A stagnant layer or lid of highly viscous fluid develops in the uppermost coldest part of the top cold thermal boundary layer when r > rcl, where r = rcl = 1.18 x 103R,0~308 andR, is the Rayleigh number based on the viscosity at the top boundary. Three-dimensional convection occurs in a rectangular pattern beneath this stagnant lid. The planform consists of hot upwelling plumes at or near the centre of a rectangle, sheets of cold sinking fluid on the four sides, and cold sinking plume concentrations immersed in the sheets. A stagnant lid does not develop, i.e. convection involves all of the fluid in the box when r rc2 = 3.84 x 106R;1.36. The planform of the convection is rectangular with the coldest parts of the sinking fluid and the hottest part of the upwelling fluid occurring as plumes at the four corners and at the centre of the rectangle, respectively. Both hot uprising plumes and cold sinking plumes have sheet-like extensions, which become more well-developed as r increases. The whole-layer mode of convection occurs as two-dimensional rolls when r < min (rcl, rc2). The Nusselt number Nu depends on the viscosity at the top surface more strongly in the regime of whole-layer convection than in the regime of stagnant-lid convection. In the whole-layer convective regime, Nu depends more strongly on the viscosity at the top surface than on the viscosity at the bottom boundary.

3‐D Convection With Variable Viscosity

Abstract: Accepted 1990 September 6. Received 1990 July 16; in original form 1990 May 3 SUMMARY report numerical calculations for 3-D convection with variable viscosity. A hybrid spectral and finite difference method is used. The coupling of modes in the equation of motion, which is caused by lateral viscosity variations, is treated iteratively. Solutions for bimodal, hexagonal, square, triangular and spoke patterns are reported for bottom heated convection at infinite Prandtl number. The Rayleigh number, based on the viscosity at the mean of top and bottom temperature, is between critical and lo5, and temperature-induced viscosity contrasts up to 100 are considered (lo00 in one case). In agreement with results from laboratory experiments we find that at low Rayleigh number temperature-dependent viscosity favours flow patterns like squares or hexagons, where a columnar rising current is surrounded by sheet-like descending flow. The dichotomy in geometry between upwelling and sinking flow becomes more pronounced with increasing viscosity contrast. The temperature dependence of viscosity gives rise to a toroidal velocity component; however, it amounts only to a few per cent of the total velocity. In contrast, at the earth’s surface an approximate equipartitioning of poloidal and toroidal energy is found. We show that with non-Newtonian and depth-dependent rheology the toroidal component at the free surface can become significant, and a pattern reminiscent of plate motion can arise in a free convection model. Although these results are obtained in a parameter range which is not directly applicable to the earth, they support the conclusions that (i) upwelling flow in the mantle is unlikely to be sheet-like and will probably be in the form of columnar plumes, and that (ii) the toroidal motion found at the earth’s surface is due to the highly non-linear rheology which leads to the existence of mobile surface plates and is not caused by viscosity variations related to lateral temperature contrasts deeper in the mantle.

A Prediction of the Multifractal Model: the Intermediate Dissipation Range

Abstract: It is shown that the multifractal model of fully developed turbulence predicts a new form of universality for the energy spectrum E(k), which can be tested experimentally. Denoting by R the Reynolds number, log E/logR should be a universal function of log k/log R. This includes an intermediate dissipation range in which a continuous range of multifractal scaling exponents are successively turned off by viscosity.

Experimental study of directional solidification of aqueous ammonium chloride solution

Abstract: Directional solidification experiments have been carried out using the analog casting system of NH4Cl-H2O solution by cooling it from below with a constant-temperature surface ranging from -31.5 C to +11.9 C. The NH4Cl concentration was 26 percent in all solutions, with a liquidus temperature of 15 C. It was found that finger convection occurred in the fluid region just above the mushy layer in all experiments. Plume convection with associated chimneys in the mush occurred in experiments with bottom temperatures as high as +11.0 C. However, when the bottom temperature was raised to +11.9 C, no plume convection was observed, although finger convection continued as usual. A method has been devised to determine the porosity of the mush by computed tomography. Using the mean value of the porosity across the mush layer and the permeability calculated by the Kozeny-Carman relationship, the critical solute Rayleigh number across the mush layer for onset of plume convection was estimated to be between 200 and 250.

Non-Darcy Mixed Convection Along a Vertical Wall in a Saturated Porous Medium

Abstract: In most of the previous studies of either natural or mixed convection, the boundary-layer formulation of Darcy's law and the energy equation were used. However, the inertial effect is expected to become very significant when the pore Reynolds number is large. This is especially true for the case of either the high Rayleigh number regime or for high-porosity media. In spite of its importance in many applications, the non-Darcy flow effect has not received much attention. In this note, non-Darcy flow effects, which include the inertial and thermal dispersion effects, are closely examined. Steady-state non-Darcy convection, in the form of natural, mixed, and forced convection, is considered for a heated vertical surface embedded in a saturated porous medium.

Chaotic advection in a Rayleigh-Bénard flow

Abstract: We consider the problem of transport of a passive tracer in the time-dependent flow corresponding to a Rayleigh number scrR slightly above the scrRt at the onset of the even oscillatory instability for Rayleigh-Benard convection rolls. By modeling the flow with a stream function, we show how to construct and identify invariant structures in the flow that act as a ‘‘template’’ for the motion of fluid particles, in the absence of molecular diffusivity. This approach and symmetry considerations allow us to write explicit formulas that describe the tracer transport for finite times. In the limit of small amplitude of the oscillation, i.e., when (scrR-scrRt)1/2 is small, we show that the amount of fluid transported across a roll boundary grows linearly with the amplitude, in agreement with the experimental and numerical findings of Solomon and Gollub [Phys. Rev. A 38, 6280 (1988)]. The presence of molecular diffusivity introduces a (long) time scale into the problem. We discuss the applicability of the theory in this situation, by introducing a simple rule for determining when the effects of diffusivity are negligible, and perform numerical simulations of the flow in this case to provide an example.

Linear stability analysis of Bénard–Marangoni convection in fluids with a deformable free surface

Abstract: Linear stability analysis of the Benard–Marangoni problem in a layer of fluid with a deformable free surface is considered. The analysis is restricted to fixed values of the Prandtl and the Biot numbers in order to determine the role of the Crispation number on convection. For a deformable upper interface both stationary and oscillatory instabilities are obtained. These two kinds of instabilities have been studied separately and the corresponding critical wave numbers kc and critical Rayleigh numbers Rc have been obtained numerically. The conditions under which two stationary states, an overstable mode and stationary mode, or two overstable modes can coexist simultaneously are determined. In the last case the possibility to obtain a strong resonance between two overstable modes is also discussed.

Asymmetric modes and the transition to vortex structures in rotating Rayleigh-Bénard convection

Abstract: We report heat transport measurements and optical shadowgraph visualization of rotating Rayleigh-B\'enard convection. For dimensionless rotation rates 1404300, the initial transition to convection, occurring at a Rayleigh number R much less than the linear-stability value for roll or vortex states, is a forward Hopf bifurcation to an azimuthally asymmetric state with mode number n. States with n=3, 4, 5, 6, and 7 exist at low to moderate R and precess with frequencies that depend on R and \ensuremath{\Omega}. At higher R there is a continuous transition to a state with noisy, time-dependent heat transport, a distinct array of vortices in the central region, and a modulation of the precession speed of the outer structures.

A computational study of Rayleigh–Bénard convection. Part 1. Rayleigh-number scaling

Abstract: A parametric study is made of chaotic Rayleigh-Benard convection over moderate Rayleigh numbers. As a basis for comparison over the Rayleigh number (Ra) range we consider mean quantities, r.m.s. fluctuations, Reynolds number, probability distributions and power spectra. As a further means of investigating the flow we use the Karhunen-Loeve procedure

Experimental verification of Darcy-Brinkman-Forchheimer flow model for natural convection in porous media

Abstract: Experimental results for natural convection in a horizontal porous cavity of aspect ratio A = 5 and heated from below are reported. A wide range of governing parameters are covered by careful selection of bead size, solid material, and fluid. These results fully support the effects of fluid-flow parameters (Rayleigh and Prandtl numbers), porous matrix-structure parameters (Darcy and Forchheimer numbers), and the conductivity ratio as predicted by the formulation based on the Darcy-Brinkman-Forchheimer (DBF) equations of motion. The DBF flow model, with variable porosity and variable thermal conductivity in the wall regions, predicts reasonably well in comparison with the experimental data. However, the difference between the predictions and the measurements increases as the ratio of solid-to-fluid thermal conductivity becomes very large. 32 refs.

The Ra-Pr domain of laminar natural convection in an enclosure heated from the side

Abstract: The phenomenon of natural convection in a square enclosure heated and cooled in the horizontal direction was investigated numerically in the Prandtl number range 0.01-10 and the Rayleigh number range 102-1011. The numerical method relied on the full governing equations for time-dependent flows. The study focused on the detection of inertia-sustained fluctuations in the flow field and on the highest Rayleigh number where steady-state laminar flows are possible. It was found that the highest Rayleigh number decreases dramatically as the Prandtl number decreases. This finding agrees qualitatively with experimental observations of transition to turbulent natural convection and with the “local Reynolds number” criterion of transition to turbulence recommended by the buckling theory of turbulent flow.

Instability of stationary unbounded stratified fluid

Abstract: Suppose that the density of stationary unbounded viscous fluid is a sinusoidal function of the vertical position coordinate z. Is this body of fluid gravitationally unstable to small disturbances, and, if so, under what conditions, and to what type of disturbance? These questions are considered herein, and the answers are that the fluid is indeed unstable, for any non-zero value of the amplitude of the sine wave, to disturbances with large horizontal wavelength. These disturbances have approximately vertical velocity everywhere and tilt the alternate layers of heavier and of lighter fluid, causing the fluid in the former to slide down and that in the latter to slide up, leading to a sinusoidal variation of the vertically averaged density and thereby to reinforcement of the vertical motion. The identification of this novel and efficient global instability mechanism prompts a consideration of the stability of other cases of unbounded fluid stratified in layers. Two other types of undisturbed density distribution, the first an isolated central layer of heavier or lighter fluid, with density varying say as a Gaussian function, and the second an isolated layer of fluid in which the density varies as the derivative of a Gaussian function, are found to be unstable, at all values of the magnitude of the density variation, to disturbances having the same global character. For the first of these two types of density distribution, the behaviour of a disturbance with long horizontal wavelength depends only on the net excess mass of unit area of the central layer, and for the second it depends only on the first moment of the density in the central layer. For the second type there arises another global instability mechanism in which light fluid is stripped away from one side of the layer and heavy fluid from the other without any tilting. In all cases the properties of a neutral disturbance are determined numerically, and the growth rate is found as a function of the Rayleigh number, the Prandtl number, and the horizontal wavenumber of the disturbance. An energy argument gives results easily for the inviscid non-diffusive limit, when all disturbances grow, and reveals the tilting-sliding mechanism of the instability of a disturbance with large horizontal wavelength in its simplest form.

Collisions between pulses of traveling-wave convection.

Abstract: The first nonlinear state of traveling-wave convection in binary fluids with moderate negative separation ratio in a narrow geometry consists of ``pulses'': localized patches of convection whose spatial shape is fixed, and that drift at a velocity that depends on the local Rayleigh number. I present the results of two kinds of experiments on traveling-wave pulses. First, I study the behavior of pulses as they drift past narrow, fixed peaks in Rayleigh number. The pulses are sensitive to the Rayleigh number in a spatial domain that extends far ahead of the main body of the pulse. Second, I study collisions between pairs of counterpropagating pulses as a function of the velocity with which they approach one another. At high approach velocity, only one pulse survives the collision. At low approach velocity, a persistent double-peaked structure is formed. Under certain circumstances, this structure can be interpreted as a weakly bound state of two pulses.

Exact solitary waves in a convecting fluid

Abstract: The perturbed Korteweg-de Vries equation ut+ lambda 1uux+ lambda 2uxxx+ lambda 3uxxxx+ lambda 4 (uux)x+ lambda 5uxx=0, which describes the evolution of long shallow waves in a convecting fluid when the critical Rayleigh number slightly exceeds its critical value, admits two types of exact solitary wave solution.

A two‐phase, two‐component model for natural convection in a porous medium

Abstract: A numerical study of natural convection in a two-phase, two-component flow in a porous medium heated from below is presented. Interphase mass and energy transfer, latent heat and bouyancy effects are major physical features. This study extends earlier studies of natural convection based on single-phase, saturated porous medium models. The appearance of two-phase heat pipe zones in the flow has a marked effect on the fluid and heat flows as well as on the performance of the numerical methods. The numerical techniques for handling phase change, Jacobian construction and time step selection are discussed.

Active control of convection

Abstract: It is demonstrated theoretically that active (feedback) control can be used to alter the characteristics of thermal convection in a toroidal, vertical loop heated from below and cooled from above. As the temperature difference between the heated and cooled sections of the loop increases, the flow in the uncontrolled loop changes from no motion to steady, time‐independent motion to temporally oscillatory, chaotic motion. With the use of a feedback controller effecting small perturbations in the boundary conditions, one can maintain the no‐motion state at significantly higher temperature differences than the critical one corresponding to the onset of convection in the uncontrolled system. Alternatively, one can maintain steady, time‐independent flow under conditions in which the flow would otherwise be chaotic. That is, the controller can be used to suppress chaos. Likewise, it is possible to stabilize periodic nonstable orbits that exist in the chaotic regime of the uncontrolled system. Finally, the controller also can be used to induce chaos in otherwise laminar (fully predictable), nonchaotic flow.

Natural convection above an array of open cavities heated from below

Abstract: The effect of buoyancy on the flow and heat transfer that develop between a horizontal cold surface and an infinite two-dimensional array of open cavities heated from below is studied numerically. In earlier investigations the steady-state features of this problem were studied for the case of unbounded flow above the cavities. The resulting flow pattern was found to be symmetrical with respect to the centerlines of the cavities. In the present work it is shown that the symmetry of the flow can be destroyed due to the presence of an upper wall. The evolutionary path to steady-state flow is examined, and sustained oscillatory behavior has been observed in several cases. The solution structure is governed by five parameters, i.e., the geometric parameters A = l'/H', B = h'/H', and C = L'/H', the Rayleigh number Ra = gβ ΔT' H' 3/av, and the Prandtl number Pr = v/α. For a geometry with A = ½z, B = ¼, and C = 1, a complicated solution structure is observed upon increasing the Rayleigh number. For Ra ≤ ...

Throughflow effects on convective instability in superposed fluid and porous layers

Abstract: We implement a linear stability analysis of the convective instability in superposed horizontal fluid and porous layers with throughflow in the vertical direction. It is found that in such a physical configuration both stabilizing and destabilizing factors due to vertical throughflow can be enhanced so that a more precise control of the buoyantly driven instability in either a fluid or a porous layer is possible. For ζ = 0.1 (ζ, the depth ratio, defined as the ratio of the fluid-layer depth to the porous-layer depth), the onset of convection occurs in both fluid and porous layers, the relation between the critical Rayleigh number Rcm and the throughflow strength γm is linear and the Prandtl-number (Prm) effect is insignificant. For ζ ≥ 0.2, the onset of convection is largely confined to the fluid layer, and the relation becomes Rcm ∼ γ2m for most of the cases considered except for Prm = 0.1 with large positive γm where the relation Rcm ∼ γ3m holds. The destabilizing mechanisms proposed by Nield (1987 a, b) due to throughflow are confirmed by the numerical results if considered from the viewpoint of the whole system. Nevertheless, from the viewpoint of each single layer, a different explanation can be obtained.

Correlations for Natural Convection Between Heated Vertical Plates

Abstract: This paper presents the numerical results of natural convective flows between two vertical, parallel plates within a large enclosure A parametric study has been conducted for various Prandtl numbers and channel aspect ratios The results are in good agreement with the reported results in the literature for air for large aspect ratios However, for small aspect ratios, the present numerical results do not agree with the correlations given in the literature The discrepancy is due to the fact that the published results were obtained for channels where the diffusion of thermal energy in the vertical direction is negligible The results obtained in this paper indicate that vertical conduction should be considered for channel aspect ratios less than 10 for Pr = 07 Correlations are presented to predict the maximum temperature and the average Nusselt number on the plate as explicit functions of the channel Rayleigh number and the channel aspect ratio for air The plate temperature is a weak function of Prandtl number for Prandtl numbers greater than 07, if the channel Rayleigh number is chosen as the correlating parameter For Prandtl numbers less than 01, the plate temperature is a function of channel Rayleigh number and the Prandtl numbermore » A correlation for maximum temperature on the plate is presented to include the Prandtl number effect for large aspect ratio channels« less