Supercritical Rayleigh Numbers Research Articles

Active control of Rayleigh–Bénard convection

We report on stabilizing the unstable no-motion state in a moderate aspect ratio one-dimensional Rayleigh–Bénard convection experiment. A linear proportional control algorithm uses shadowgraphic convection images to determine heat flux perturbations which are applied to the lower boundary by a network of local heaters. We show that simple linear control stabilizes the otherwise unstable no-motion (conduction) state over a substantial range of supercritical Rayleigh numbers.

Temporal, spatial and thermal features of 3-D Rayleigh-Bénard convection by a least-squares finite element method

Numerical solutions of 3-D time-dependent Rayleigh-Bénard convection are presented in this work. The temporal, spatial and thermal features of convective patterns are studied for four different geometric aspect ratios, 2:1:2, 4:1:4, 5:1:5 and 3.5:1:2.1 at supercritical Rayleigh numbers Ra = 8 × 10 3, 2.4 × 10 4 and Prandtl numbers Pr = 0.71, 2.5. Several physical phenomena, such as multicellular flow pattern, oscillatory transient solution, ‘T-shaped’ rolls at the ends of a rectangular box, and roll alignment, are observed in our simulations. The numerical technique is based on an implicit, fully coupled, and time-accurate method, which consists of the Crank-Nicolson scheme for time integration, Newton's method for the convective terms with extensive linearization steps, and a least-squares finite element method. A matrix-free algorithm of the Jacobi conjugate gradient method is implemented to solve the symmetric, positive definite linear system of equations.

Penetrative convection induced by the freezing of seawater

Thermosolutal convection induced by freezing a layer of seawater from above is studied in a series of numerical experiments. The model equations take into account the dependence of the density and the temperature of maximum density on salt content, the possible presence of penetrative convection and supercooling effects. The threshold values for the onset of convection are determined over a wide range of parameter space. Two types of cell pattern are predicted by the linear theory: (1) convection of the entire fluid cell and (2) multi-layer convection. An attempt is made at quantifying the cabbeling instability associated with the onset of the latter. Only multi-layer convection prevails for supercritical Rayleigh numbers. For a specific parameter range, a flow pattern consisting of a strongly convecting layer floating between two relatively stable layers is found. The numerical experiments seem to imply that penetrative convection delays the release of heat to the atmosphere during the formation of polynyas.

Convection in cylindrical containers at supercritical Rayleigh numbers

An experimental study is described of various cellular convection patterns in cylindrical containers at supercritical Rayleigh numbers. The fluid containers are heated from below and as the Rayleigh number is slowly increased the convection patterns undergo a series of bifurcations to nonaxisymmetric configurations. The form of convection depends on both the Rayleigh number and aspect ratio of the container. The observations are discussed in the light of current understanding of pattern-forming phenomena.

NUMERICAL SIMULATION OF FLOW PATTERN FORMATION IN A BOTTOM HEATED CYLINDRICAL FLUID LAYER

The temporal formation of the buoyancy-driven flow structures in a bottom heated, shallow, cylindrical fluid layer was numerically studied. The unsteady three-dimensional Navier-Stokes and energy equations were discretized by the power law scheme and solved by the fully implicit Marker-and-Cell method. Computations were carried out for the pressurized argon (Pr = 0.69) and water (Pr = 6.1) layers for various Rayleigh numbers and heating rates of the layer. In the pressurized argon layer at a slightly supercritical Rayleigh number with Ra{sub f} = 1.05Ra{sub c} a steady straight roll pattern was formed when the heating rate was very slow (a = 0.001) after a long transient stage. When the heating rate was raised to a = 0.01, a very different structure like U-rolls was formed at steady state. In the water layer with Ra{sub f} = 1.05Ra{sub c}, a straight roll pattern was again formed, but at a = 0.01. At Ra{sub f} = 1.13Ra{sub c}, curved rolls with the three foci at the sidewall were formed for a = 0.01. A pattern in the form of U-rolls appears at a = 0.01. Regular concentric circular rolls prevail at a = 1.0. When the Rayleigh number is further raised tomore » 1.23Ra{sub c}, the resulting steady flow is dominated by incomplete circular rolls with open ends near {theta} = 0{degree}.« less

Laminar, buoyancy induced flow structures in a bottom heated, aspect ratio 2 duct with throughflow

Three-dimensional, transient calculations have been made for flow far downstream in a horizontal, bottom heated, aspect ratio 2 duct with throughflow in the low supercritical Rayleigh number ( Ra) and low Reynolds number ( Re) regime. The two flow types present are longitudinal rolls and distorted, convecting transverse rolls. Longitudinal rolls results confirm previous studies. Ra and Re dependence of the transverse rolls flow field is investigated. The Nusselt number ( Nu) is seen to be essentially independent of Re, and perhaps also of the flow regime. Isotherms and velocity vector plots are presented. Comparisons are made with published experimental and numerical results.

Influence of the heating rate on supercritical Rayleigh-Bénard convection

The time dynamics of Rayleigh-Bénard convection originating from different heating rates for a Boussinesq fluid of Prandtl number Pr = 0.71 (air) has been studied numerically inside a three-dimensional (3D) rectangular box of aspect ratios γ z = 1.25, γ y = 2. The heating rate, introduced through a time dependent Rayleigh number R( t), drives a flow transition at least in the range of supercritical Rayleigh numbers ( R s ) here considered to solve the governing equations; R s = 3.6 x 10 3, 5 × 10 3, 9 × 10 3, 1.3 × 10 4 and 1.6 × 10 4. The flow transition, identified by a change of the rotation sense of a two-roll fluid pattern was found when perfectly conducting side walls were used. This kind of flow transition has been reported in recent experiments.

Three-dimensional convection in rectangular domains with horizontal throughflow

The Rayleigh-Bénard convection with a superimposed horizontal shear flow in rectangular domains is treated numerically utilizing a finite difference method and experimentally by applying an interferometric measuring technique. The fluid used is silicon oil, a high Prandtl number fluid, Pr = 530. The investigations concern the supercritical Rayleigh number region of 2300 ⩽ Ra ⩽ 20 000. The Reynolds number, which characterizes the strength of the throughflow, varies in the range of 3 × 10 −5 ⩽ Re ⩽ 0.1. According to special parameter combinations of the Rayleigh and the Reynolds number, convective rolls with different orientations can be distinguished. The dynamical behavior of travelling, transverse rolls is also considered.

Convection in a porous medium with solidification

The coupled effects of thermal convection and solidification of a Single-component liquid in a porous medium are investigated. A rigorous two-parameter perturbation analysis is used to determine the effects on both the stability of the basic state of heat conduction and the stability of finite-amplitude convection. The analysis shows that due to the kinematic conditions at the solid/liquid interface, hexagons having upflow in the center are stable near the onset of convection. For sufficiently supercritical Rayleigh numbers, however, rolls are the only stable mode. The transition from hexagons to rolls is characterized by a hysteresis loop. Moreover, the transition is shown to be controlled by one particular critical value of the convection amplitude. This generic property holds for non-Boussinesq convection in bulk liquid-layers too.

Spatial and thermal features of three dimensional Rayleigh-Bénard convection

A full 3-D numerical study is reported on natural convection of air in six heated-from-below parallelepipedic enclosures. A supercritical Rayleigh number (Ra = 8 × 10 3) , was used in order to track the evolution of convective structures as a function of the enclosure's aspect ratio ( A) only. The six situations (1 ⩽ A ⩽ 5) exhibit a characteristic toroidal flow pattern which can be seen, at the lowest aspect ratio, as a unicellular structure which evolves to a multicellular combination of concentric roll-cells at higher aspect ratios. The transition from one convective structure to another has the features of a flow bifurcation controlled by A. The overall Nusselt number changed continuously when the aspect ratio was increased. At the lowest aspect ratios the change was significant, but further increments in A did not produce important variations in this parameter, in the range of aspect ratios studied.

The effect of a weak heterogeneity of a porous medium on natural convection

The results of an investigation on the effect of a weak heterogeneity of a porous medium on natural convection are presented. A medium heterogeneity is represented by spatial variations of the permeability and of the effective thermal conductivity. As a general rule the existence of horizontal thermal gradients in heterogeneous porous media provides a sufficient condition for the occurrence of natural convection. The implications of this condition are investigated for horizontal layers or rectangular domains subject to isothermal top and bottom boundary conditions. Results lead to a restriction on the classes of thermal conductivity functions which allow a motionless solution. Analytical solutions for rectangular weak heterogeneous porous domains heated from below, consistent with a basic motionless solution, are obtained by applying the weak nonlinear theory. The amplitude of the convection is obtained from an ordinary non-homogeneous differential equation, with a forcing term representative of the medium heterogeneity with respect to the effective thermal conductivity. A smooth transition through the critical Rayleigh number is obtained, thus removing a bifurcation which usually appears in homogeneous domains with perfect boundaries, at the critical value of the Rayleigh number. Within a certain range of slightly supercritical Rayleigh numbers, a symmetric thermal conductivity function is shown to reinforce a symmetrical flow while antisymmetric functions favour an antisymmetric flow. Except for the higher-order solutions, the weak heterogeneity with respect to permeability plays a relatively passive role and does not affect the solutions at the leading order. In contrast, the weak heterogeneity with respect to the effective thermal conductivity does have a significant effect on the resulting flow pattern.

A study of hydrothermal convection in saturated porous media

Abstract Because of its relevance to many geological and technical problems, hydrothermal convection is investigated here mainly with the aid of numerical models by a systematic analysis of the properties of this type of convection for a range of super-critical Rayleigh numbers. Calculations were performed for two-dimensional models with constant properties in a region of aspect ratio 2. The principal results in the case of temperature fixed at the impermeable top and bottom are the following for the Nusselt number Nu, the cell aspect ratio a, and the boundary layer thickness δ: Nu ≈ 1.7 R0.5, a ≈ 1.3 R−0.4, δc ≈ 0.4 R−0.4 for 2.5 R = R f /R f ∗ and Rf and R f ∗ are the ambient and critical filtration Rayleigh numbers, respectively, relevant to the hydrothermal problem. Beside the usual critical Rayleigh number R f ∗ two additional, new ones have been found: R f ∗∗ which characterizes the appearance of boundary layers, and R f ∗∗∗ which marks the transition to the non-steady state. A number of definitions of the boudary layer are analyzed, and a new definition is proposed based on the condition that the conductive heat flux is equal or greater than the convective one. The initial oscillations after the onset of convection in our model are analyzed and interpreted for the flow in porous media in comparison to the viscous case. Finally a diagram is constructed for the easy evaluation of the numerical accuracy as a function of grid size.

The effect of imperfectly insulated sidewalls on natural convection in porous media

The paper presents the results of an investigation of the effect of imperfectly insulated sidewalls on natural convection in porous media at slightly supercritical Rayleigh numbers. An analytical solution for a rectangular domain with imperfectly insulated sidewalls and heated from below, was obtained through the weak nonlinear theory. The solution enables the determination of the amplitude of the convection and the direction of the flow. The amplitude results from an ordinary nonhomogeneous differential equation, with a forcing term representing the heat leakage through the lateral walls. The steady state amplitude solution shows that the transition through the critical Rayleigh number is smooth, differing from the case of perfectly insulated sidewalls where a bifurcation usually appears at the critical Rayleigh number. As a result, within a certain range of slightly supercritical Rayleigh number values, the amplitude and the direction of the convection currents are uniquely determined by the heat leakage through the lateral walls and they are independent of the initial conditions. A subcritical convection occurs as a result of the imperfectly insulated sidewalls, enabling the smooth transition through the critical Rayleigh value. A three-branch bifurcation develops at a higher Rayleigh number. A stability analysis of the solutions, corresponding to these branches, shows that the amplitudes which correspond to the two highest values are stable, while the third is unstable.

Natural convection in a horizontal porous cylinder

Natural convection in a porous horizontal circular cylinder is considered. The cylinder wall is non-uniformly heated to establish a linear temperature in the vertical direction, with the end sections perfectly insulated. When L > 0.86, a unique three-dimensional flow is determined at the onset of convection. For short cylinders (L < 0.86) the convection is two-dimensional. In this case there exist two different steady solutions at supercritical Rayleigh numbers, consisting of two and three rolls, respectively. It is proved that both flow structures and any composition of these structures are stable. However, introducing thermal forcing in the applied temperature the flow becomes uniquely determined.

Rayleigh convection, mass transport, and change in porosity in layers of sandstone

The critical Rayleigh number for the onset of thermal convection in a quartz‐rich sandstone layer is discussed for different thermal boundary conditions. The mass transfer of silica due to Rayleigh convection is computed, assuming that the solution is saturated with respect to quartz and that the silica concentration is a function of temperature only. Since quartz solubility increases with temperature, silica is dissolved from the sandstones in the lower part of the layer where the fluid is being heated and precipitated in the upper part where it is cooled. The change in porosity is found as function of the vertical coordinate, valid for small values of time. It is shown that for small supercritical Rayleigh numbers the time for the porosity to change up to 10% of its initial value is of the order 10 m.y. The spatial variation in porosity should be a characteristic signature of Rayleigh convection and should be easy to recognize if convection has taken place for significant geological time.

An Analytical Study on Natural Convection in Isotropic and Anisotropic Porous Channels

This paper is an analytical study on natural two-dimensional convection in horizontal rectangular channels filled by isotropic and anisotropic porous media. The channel walls, assumed to be impermeable and perfectly heat conducting, are nonuniformly heated to establish a linear temperature distribution in the vertical direction. We derive the critical Rayleigh numbers for the onset of convection and examine the steady flow patterns at moderately supercritical Rayleigh numbers. The stability properties of these flow patterns are examined against two-dimensional perturbations using a weakly nonlinear theory.

Direct numerical simulation of thermal convection over a wavy surface

The influence of a wavy surface on thermal convection of Rayleigh-Benard type in a Boussinesq fluid is investigated by direct numerical simulations. The surface height varies sinusoidally in one direction. The wave amplitude amounts up to 10% of the fluid layer height and the wavelength equals about the critical wavelength of Rayleigh-Benard convection. The horizontal size of the computational domain equals this wavelength. For isothermal no-slip boundaries, two-dimensional convection sets in at subcritical Rayleigh numbers in close agreement with linear theory. The heat-transfer rate grows almost with the square of the surface-wave amplitude. Convection in a fluid layer over a no-slip surface with prescribed heat flux and an adiabatic free-slip boundary at the top is investigated for supercritical Rayleigh numbers and a Prandtl number of 0.7 in two and three dimensions. Two-dimensional simulations show oscillatory roll convection which becomes almost stationary if the Rayleigh number is of order 7000 or less. The two-dimensional convection is unstable with respect to three-dimensional disturbances and a cross-role pattern evolves even over a surface which is undulated in one direction only. For Rayleigh numbers exceeding about 15 000, the flow becomes turbulent. The results exhibit little sensitivity of the convection to the wavy surface for a 10% amplitude.

Molecular dynamics versus hydrodynamics in a two-dimensional Rayleigh-Bénard system.

We compare a microscopic simulation of a fluid made up of 5000 hard disks and maintained at a supercritical Rayleigh number to the corresponding macroscopic hydrodynamics. Very good quantitative agreement is demonstrated.

Convection experiments in high Prandtl number silicones, Part 1. Rheology, equipment, nomograms and dynamic scaling of stress- and temperature-dependent convection in a centrifuge

It has been impossible to model mantle convection by other than numerical methods, because dynamically scaled laboratory experiments (of practical size) in high Prandtl number fluids, would require a temperature gradient reaching far above the sublimation temperature of any known fluid. However, convection can be established in high Prandtl number fluids without large temperature gradients and length scales if the thermal buoyancy forces are enhanced by increasing the gravitational acceleration. This has been realized by placing the test tank in a high speed centrifuge (up to 3000 rpm) at The Hans Ramberg Tectonic Laboratory (Uppsala); the centrifugal acceleration imposes an apparent gravity field several thousand times the Earth's gravity. By using this method, supercritical Rayleigh numbers up to 10 6 can be achieved in a transparent polymer fluid (SGM36) of Prandtl number 10 8. The carefully developed test tank and control unit, and a rigorous derivation of the conditions for dynamic similarity of convection in fluids with temperature- and stress-dependent rheology facilitates detailed laboratory studies of high Prandtl number convection. The working philosophy (section 2), equipment (section 3), non-Newtonian and temperature-dependent scaling procedure (section 4), and practical nomograms (section 5) are presented here to aid such laboratory studies. The significance of this new laboratory method for studying convection is outlined in section 6. Real convection experiments in high viscosity polydimethylsiloxanes now have the potential to contribute to the understanding of six major aspects of convection. These are: 1. (1) progressive deformation and displacement in three-dimensional convective flow fields, 2. (2) convective mixing, 3. (3) convection with high Prandtl numbers, 4. (4) non-Newtonian flow, 5. (5) temperature-dependent flow, and 6. (6) high Rayleigh number convection. Detailed applications will be discussed in a companion paper (Part 2 of the present project).

Three-dimensional thermal cellular convection in rectangular boxes

The extension of the classic Rayleigh–Bénard problem of a horizontal layer heated from below to the three-dimensional convection in rectangular boxes is dealt with in this paper both numerically and experimentally. Also discussed is the influence of shear flows in tilted boxes and the transition to time-dependent oscillatory convection. Three-dimensional numerical simulations allow the calculation of stationary solutions and the direct simulation of oscillatory instabilities. We limited ourselves to laminar and transcritical flows. For studying the particular characteristics of three-dimensional convection in horizontal containers, we carefully selected two container geometries with aspect ratios of 10:4:1 and 4:2:1. The onset of steady cellular convection in tilted boxes is calculated by an iterative application of a combined finite-difference method and a Galerkin method. The appearance of longitudinal and transverse convection rolls is determined by means of inter-ferometrical measuring techniques and is compared with the results of the linear stability theory. The spatial flow structure and the transition to oscillatory convection is calculated for selected examples in the range of supercritical Rayleigh numbers. Experimental investigations concerning the stability behaviour of the steady solutions with regard to time-dependent disturbances show a distinct influence of the Prandtl number and confirm the importance of nonlinear effects.