High Rayleigh Number Convection Research Articles

Correspondence principle for turbulence: Application to the Chicago experiments on high Rayleigh number Bénard convection

The correspondence principle postulated for the description of hydrodynamic turbulence [Phys. Rev. Lett. 57, 1722 (1986)] combined with the theory of thermal boundary layer [B. Castaing et al. (private communication)] is applied to high Rayleigh number convection in a Bénard cell. Quantitative interpretation of recent experimental data [B. Castaing et al. (private communication)] is presented. The predicted intermittency exponent following from comparison of the theory with experiment is 0.175<μ<0.275. A crucial experimental test of the renormalization group theory of turbulence is proposed.

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).

Mass and heat transfer by high Rayleigh number convection in a porous medium heated from below

This paper outlines a combined theoretical and numerical study of the mass transfer effected by high Rayleigh number Bénard convection in a two-dimensional saturated porous layer heated from below. The focus of this study is on the Darcy flow, heat transfer and mass transfer scales of a single cell (roll) that exists in the steady two-dimensional convection regime. The numerical solutions are based on the complete governing equations for two-dimensional flow, and cover the Rayleigh number range 50–2000. The numerical results compare favorably with the theoretical conclusions of a scale analysis that is based on the recognition of 1. (i) two temperature difference scales in the cell, 2. (ii) a flow field without horizontal boundary layers, and 3. (iii) thermal top and bottom end-regions that are not slender enough to be boundary layers. Writing Le for the Lewis number, the overall mass transfer rate or Sherwood number is shown to scale as Le 1 2 Ra 7 8 if Le > Ra 1 4 , as Le 2 Ra 1 2 if Ra − 1 4 < Le < Ra 1 4 , and as O(1) if Le < Ra − 1 4 . The transition from the Darcy flow to the inertia-dominated Forschheimer flow and the scales of the Forschheimer regime are discussed in the closing section.

High Rayleigh number convection in a fluid overlaying a porous bed

A series of numerical experiments concerning the phenomenon of natural convection in a two-layer composite system heated from below is reported. The composite system consists of a fluid layer bounded from below by a porous bed saturated by the same fluid. The numerical solutions are based on the full equations for time-dependent flow, and document the main features of the flow at Rayleigh numbers several orders of magnitude above critical value. The ranges covered by the numerical solutions are 10 2–10 6 for overall Rayleigh number, 10 −7–10 −4 for Darcy number, and 0.2–1 for the height to length ratio of the composite system

Numerical study of high Rayleigh number convection in a medium with depth-dependent viscosity

The equations of motion are solved numerically for a Boussinesq fluid with infinite Prandtl number in a square 2-D box where the viscosity increases with depth. Three heating modes are employed: bottom heating, internal heating, and half bottom and half internal heating. In all cases the boundaries are free slip. The range of Rayleigh numbers employed is 10^4-10^7. The viscosity increases as 10^(β(1-y)), where y is distance measured from the bottom upwards and β is a free parameter. In the bottom heated cases, the convective velocities slow near the bottom and result in a large temperature drop between the bottom boundary and interior compared with the top boundary and the interior. This results in increased buoyancy in the ascending limb. In the internally heated case, the flow in the top half of the box resembles Rayleigh-Benard convection and in the bottom half it approaches a conductive thermal regime for β greater than about 2. In this case the top surface heat flux decays from ascending to descending limb and the ascending and descending limbs become more equal in their buoyancy. Increasing β decreases the efficiency of heat transport, but has little effect on the exponents of Nu-Ra and Pe-Ra relations. There is a larger decrease in heat transport efficiency for a given β in the bottom heated case compared to the internally heated case.

High Rayleigh number convection with strongly variable viscosity: A comparison between mean field and two‐dimensional solutions

We have studied single‐mode mean field equations associated with variable viscosity convection for both steady state and time‐dependent situations. Steady state mean field solutions can be obtained by treating the governing mean field equations as two coupled fourth‐order differential systems, which are solved as a series of coupled two‐point boundary value problems with underrelaxation. Steady solutions can be achieved in which there can exist viscosity contrasts exceeding 108 and interior Rayleigh numbers of 0(109). For comparison with two‐dimensional solutions we have employed results derived from two finite element methods. Both temperature‐dependent and temperature‐ and pressure‐dependent viscosity with the surface viscosity fixed have been studied. Our results show that the differences between the two approaches grow with both viscosity contrast and convective vigor. The steady state Nusselt numbers and interior temperatures are greater in the case of mean field solutions. The power law index β governing the relationship between the interior Rayleigh number and the Nusselt number is larger for the mean field. Comparison of time‐dependent solutions shows that one can monitor rather faithfully the evolution of the averaged interior temperature and surface heat flow over a long time scale with the mean field method. Initially, thermal instabilities originating from the mean field boundary layers are found to be correlated with two‐dimensional boundary layer instabilities and are much more violent in character and cause large oscillations of the surface heat flow. The time scales associated with the secular variations of the Urey number, representing the ratio of the heat production and the surface heat loss, agree well between the two approaches. These results suggest that the mean field equations may serve as an efficient vehicle for studying planetary convection with complicated physics.

The application of boundary layer theory to the convective flow of a fluid with a depth dependent viscosity

The boundary layer theory for high Rayleigh number convection has been generalized to study the convective flow structure of a fluid with a viscosity that increases exponentially with depth. This method is used to study convection in a square cell with free slip isothermal boundaries and with the top temperature less than the bottom temperature. The flow is driven by sidewall shear stresses approximating the rising and descending thermal plumes. Viscosity variations up to 10 4 across the cell are considered. Values from 10 5 to 10 9 for a Rayleigh number based on the surface viscosity are used in this work. The flow and thermal structure is found to differ from constant viscosity flows. The temperature drop across the bottom boundary layer is larger than across the top, and thus the interior temperature of the cell is less than the mean of the bottom and top temperatures as in the constant viscosity flows. The flow structure is no longer symmetric since the rising and descending plumes do not have equal strengths. The global heat transfer is less efficient than in a fluid with constant viscosity equal to the top viscosity in the present models. The fluid in the lower section of the cell moves more slowly relative to the upper section of the cell for a given viscosity contrast.

Numerical Study of Transient High Rayleigh Number Convection in an Attic-Shaped Porous Layer

Scaling arguments and numerical analysis are used to document the transient and steady-state regimes of natural convection in a triangular porous layer cooled from above (along the sloping wall). The numerical simulations are conducted in the high Rayleigh number domain, Ra = 100, 1000, where Ra is the Darcy-modified Rayleigh number based on height, H. The scale analysis predicts the existence of distinct thermal boundary layers if Ra1/2 (H/L) &gt; 1, where H/L is the height/length geometric ratio of the attic-shaped porous layer. The numerical simulations confirm the scaling results, as well as the prediction that the flow consists primarily of an elongated horizontal counterflow driven by the cold wall. In addition, the numerical solutions show the presence of a Be´nard-type instability at high enough Rayleigh numbers. For example, if H/L = 0.2, the instability is present when Ra &gt; 620; this critical Rayleigh number is found to increase as H/L increases.

Numerical Study of High Rayleigh Number Convection in a Vertical Porous Enclosure

Numerical Study of High Rayleigh Number Convection in a Vertical Porous Enclosure

High Rayleigh number convection in shallow enclosures with different end temperatures

Buoyancy-driven laminar convection in shallow horizontal enclosures (width/height ⪢1) with adiabatic horizontal walls and isothermal vertical walls of different temperatures has been investigated for very high Rayleigh numbers ( Ra ⪢ 10 6). It is found that this flow regime, which is characterized by boundary layers lining both the vertical and horizontal walls, differs qualitatively from the low- Ra flow regime, in which the horizontal boundary layers are absent. The decreasing influence of Ra on the Nusselt number and the effect of the aspect ratio are analytically shown and then numerically verified. Comparison is made with existing experimental and numerical data, and a new correlation is proposed for the Nusselt number in this flow regime. An explanation is also advanced regarding the current controversy about the power dependence of the Nusselt number on Ra in narrow vertical channels.

Convection experiments and the driving mechanism

The problem of mantle convection is discussed in light of laboratory and numerical experiments. Since Wegener's time our knowledge of earth behaviour has increased, so that we are in a better position to explain continental drift. The explanation of this phenomenon must inevitably include mantle convection. The problem has three aspects: the physical laws, the boundary conditions, and the material. We find that a likely model is high-Rayleigh number convection in a medium of temperature, pressure, and stress-dependent rheology; in this model we expect a strong and heavy upper thermal boundary layer and a fluid and light lower boundary layer. The upper one consists of the lithospheric plates which highly organize the large-scale circulation, while the lower one at the core-mantle boundary becomes unstable in the form of diapirs, plumes, or blobs in a fashion rather independent of the large-scale circulation. This model has the potential of reconciling the conflicting views of horizontal and vertical tectonics.

Effective Thermal Conductivity for Combined Radiation and Free Convection in an Optically Thick Heated Fluid Layer

The effective thermal conductivity for radiative heat transfer within an optically thick fluid layer undergoing high Rayleigh number convection is derived. This result is combined with available “pure” free-convection heat-transfer correlations to obtain closed-form analytical descriptions of the gross properties of a radiating fluid layer heated internally or from below. These simple solutions compare favorably with recent work in which the governing energy equation incorporating both turbulent heat transport and thermal radiation is solved numerically.

Fault zone controlled charging of a liquid‐dominated geothermal reservoir

A mathematical model is developed for the fault zone controlled charging of a geothermal reservoir. The model is used to describe a reservoir of finite vertical extent with an impermeable upper boundary. A quasi‐analytic theory is developed for high Rayleigh number convection of liquid in a rigid porous medium. In this approximation, liquid rises up the fault and spreads into the near regions of the reservoir isothermally. The cooling effect of the surface on the fault flow is confined to a thin layer near the surface. This layer grows with distance from the fault. In the far field of the aquifer the full depth of the reservoir is cooled by the surface. A study is made of the effect of various parameters such as mass flow rate, Rayleigh number, and fault width on the pressures, velocities, temperatures, and their gradients at different locations in the fault and in the aquifer. This analysis can be applied to geothermal systems where the thickness of the impermeable reservoir cap is quite small compared to the reservoir depth.

Power laws of high Rayleigh number convection

Previous theoretical and observational investigations have shown that vertical plumes are formed in the high Rayleigh number convection field over heated horizontal surfaces and that these plumes become unsteady and turbulent when the Rayleigh number is higher than about 20 times its critical value Rc. Based on these results, we conclude that the dissipation of kinetic energy takes place mainly in the surface boundary layer in high Rayleigh number laminar convection and mainly in the vertical plumes in turbulent convection, while the conversion of eddy potential energy into kinetic energy is accomplished mainly in the well-mixed main body of the fluid. On making use of these rather general conclusions concerning the kinetic energy generation and dissipation processes in the energy integrals, we are able to derive the well known 5/4 and 4/3 power laws of upward heat transfer by laminar and turbulent convections theoretically.

Subsolidus Convection in the Mantles of Terrestrial Planets

The role of heat transport by solid state mantle convection in determining the past and present thermal states of terrestrial planets is examined. Mantle convection models have relied on two-dimensional and axisymmetric three-dimensional numerical calculations incorporating the temperature and pressure dependence of mantle rheology and its non-Newtonian nature. Convection at high Rayleigh numbers has been investigated through theoretical scaling arguments and boundary layer theories; nevertheless, computational limits prevent modeling of the fully three-dimensional, time-dependent, very high Rayleigh number convection which probably prevails in terrestrial planets. Radar measurements of Venus, as well as Voyager exploration of the Galilean satellites, should also provide information on mantle convection.

The evolution of under-ice melt ponds, or double diffusion at the freezing point

In an experimental and theoretical study, we model a phenomenon observed in the summer Arctic, where a fresh-water layer at a temperature of 0°C floats both over a sea-water layer at its freezing point and under an ice layer. Our results show that the ice growth in this system takes place in three phases. First, because the fresh-water density decreases upon supercooling, the rapid diffusion of heat relative to salt from the fresh to the salt water causes a density inversion and thereby generates a high Rayleigh number convection in the fresh water. In this convection, supercooled water rises to the ice layer, where it nucleates into thin vertical interlocking ice crystals. When these sheets grow down to the interface, supercooling ceases. Second, the presence of the vertical ice sheets both constrains the temperatureTand salinitysto lie on the freezing curve and allows them to diffuse in the vertical. In the interfacial region, the combination of these processes generates a lateral crystal growth, which continues until a horizontal ice sheet forms. Third, because of theTandsgradients in the sea water below this ice sheet, the horizontal sheet both migrates upwards and increases in thickness. From one-dimensional theoretical models of the first two phases, we find that the heat-transfer rates are 5–10 times those calculated for classic thermal diffusion.

Comments on “High Rayleigh number convection in an enclosure—A numerical study”

First Page

High Rayleigh Number Convection in an Enclosure—A Numerical Study

Seven finite difference computations for convection in a square cavity at Rayleigh number ∼O(106) have been carried out for a variety of dynamical boundary conditions, Rayleigh numbers, and Prandtl numbers. The results indicate that various horizontal dynamical boundary conditions virtually make no difference to the main boundary-layer flow although free vertical boundaries double the velocity in the boundary layer adjacent to them. The scaled fields in the vertical boundary layers are independent of the Rayleigh number, and to a large extent, independent of the Prandtl number, when the two vertical boundary layers are distinct. The results compare reasonably well with Elder's experiments.

The temporal development of a model of high Rayleigh number convection

The evolution of a model of the flow in a layer of fluid suddenly heated from below at a Rayleigh number sufficient for a laboratory flow to ultimately become turbulent is investigated with numerical experiments. The numerical study simulates the flow by means of the mean field equations along the lines of Herring's (1963, 1964) pioneering study but uses a different finite difference technique and concentrates attention on the flow development rather than on the final statistically steady state. These solutions are compared with previous and some new simulations in two dimensions. The solutions confirm Herring's work and in addition show that the mean field equations, and in particular the weak-coupling approximations, describe the gross features of the model sufficiently well for the mean field equations to be used with reasonable confidence in evolutionary studies.