Rayleigh-Bénard convection

Rayleigh - Bénard convection is a fluid flow (thermal convection) due to a non-uniform temperature distribution in a plane horizontal fluid layer heated from below. Such flows result from the development of the convective instability, if the static vertical temperature gradient (the gradient that would be present in a motionless fluid under the same conditions) is large enough.

A horizontal layer of convecting fluid is the most comprehensively studied example of nonlinear systems exhibiting self-organization (pattern-forming systems). Rayleigh - Bénard convection, which shares a number of important properties with many other pattern-formation mechanisms, is considered the "granddaddy of canonical examples used to study pattern formation and behavior in spatially extended systems" (Newell et al 1993).

Convective motion enhances dramatically the heat transfer through the layer compared to the molecular heat conduction. The moving fluid parcels, which are agents of heat exchange, normally have velocities and effective free paths much greater than the corresponding figures for molecules. Therefore, the heat flux through the layer of convecting fluid may be several orders of magnitude higher than the heat flux due to molecular thermal conductivity.

## Contents

* 1 History
* 2 Linear analysis
* 3 Supercritical regimes
* 4 References
* 5 Further reading

## [edit] History

The role of non-uniform heating as the producer of most types of fluid motions in the Universe was first recognised in the mid-eighteenth century, nearly simultaneously by George Hadley and Mikhail Lomonosov. Well-directed studies of convection in horizontal fluid layers heated from below trace back to Bénard's experiments (Bénard 1900), in which the instability mechanism was, however, not purely thermal and was closely related to the thermocapillary effect. Lord Rayleigh (1916) was the first to consider a linear problem of the onset of thermal convection in a horizontal layer, and a more comprehensive analysis of this problem was given by Pellew and Southwell (1940). A highly extensive survey of the linear stability problems, including investigations of the effects of rotation and magnetic field on Rayleigh - Bénard convection, was presented in a classical monograph by Chandrasekhar (1961). Subsequent studies mainly dealt with nonlinear convection regimes and related pattern-formation processes. The volume of relevant publications has grown dramatically, and a number of monographs of a more or less wide scope summarize them [a concise review of many results that refer specifically to Rayleigh - Bénard convection and were obtained by the end of the 1990s can be found in Getling (1998)].

## [edit] Linear analysis

Figure 1: The neutral curve of the convective instability.

In its classical formulation, the problem of convective instability of an infinite horizontal fluid layer heated from below is treated in the framework of the Boussinesq approximation: the fluid density \\(\rho \\) is considered to be independent of the pressure (i.e., incompressibility is assumed) and to depend linearly on the temperature \\(T:\\)

\\[\tag{1}\rho -\rho _0=-\rho _0\alpha (T-T_0),\\]

where \\(\rho _0\\) is the density value for some suitably chosen reference temperature \\(T_0\\). The volumetric coefficient of thermal expansion \\(\alpha \\) is considered to be small and the material parameters of the fluid (kinematic viscosity \\(\nu \\), thermal diffusivity \\(\chi \\), and the coefficient \\(\alpha \\) itself) to vary little over the layer. Then the density variations can be neglected everywhere in the equations but in the buoyancy term, where it is multiplied by the gravitational acceleration \\({\mathbf g}\\).

The basic non-dimensional parameters controlling the regimes of convection are the Rayleigh and the Prandtl number,

\\[\tag{2}R=\frac{\alpha g\Delta Th^3}{\nu \chi },\quad P=\frac \nu \chi,\\]

where \\(h\\) is the layer thickness and \\(\Delta T\\) is the temperature difference between the bottom and top boundaries of the layer (if their temperatures are fixed; otherwise, the definition of \\(R\\) should be properly modified).

Figure 2: Schematic of two types of convection cells: (a) rolls; (b) hexagonal convection cells of the \\(l\\) and \\(g\\) types.

Assume that the equations of fluid motion, continuity, and heat transfer are linearised with respect to infinitesimal perturbations. Then, given \\(R\\) and the horizontal wavevector of the perturbation, \\(\mathbf k\\), the boundary-value problem of stability of the motionless state of the fluid yields growth rates \\(\lambda_n\\) (\\(n-1\\) being the number of nodes of the eigenfunction describing the vertical dependence of the vertical velocity component) that depend only on the wavenumber \\(k=|\mathbf k|\\) but not on the direction of \\(\mathbf k\\). The neutral curve \\(R=R_1(k)\\) in the plane \\((k,R)\\), i.e., the locus of points where the perturbations have zero growth rate \\(\lambda_1\\), is shown in Fig. 1. The region above this curve corresponds to growing perturbations, while they all decay in the region below the curve. The minimum value of \\(R_1(k)\\) is termed the critical Rayleigh number \\(R_\mathrm c\\), and the wavenumber at which this minimum is reached, \\(k_\mathrm c\\), is known as the critical wavenumber.

Thus, the layer is convectively unstable at \\(R>R_\mathrm c\\), while it is stable at \\(R<R_\mathrm c\\) and the case of \\(R=R_\mathrm c\\) corresponds to neutral, or marginal, stability. If the Rayleigh number (in other words, the static temperature gradient properly non-dimensionalised) is increased from below \\(R_\mathrm c\\), convection sets in first at \\(R=R_\mathrm c\\) as a steady flow of infinitesimal amplitude with the wavenumber \\(k=k_\mathrm c\\). The larger \\(R\\), the wider the band of wavenumbers \\(k\\) in which perturbations can grow. Nonlinear effects restrict their growth to a certain level. Within a fairly wide range of \\(R\\), a well-defined characteristic wavenumber (which depends not only on the parameters of the regime but also on the prehistory of the flow) can be noted in convective flows. The selection of this wavenumber is a very subtle issue, which has been addressed in numerous studies (see, in particular, Chapter 6 in Getling 2008).

Figure 3: Convection cells in experiments: (a) quasi-two-dimensional rolls observed by V.S. Berdnikov and V.A. Markov (bright specks are aluminum flakes used to visualise the flow), unpublished; (b) hexagonal cells in a fluid with a strong temperature dependence of viscosity (Richter 1978).

The values of \\(R_\mathrm c\\) and \\(k_\mathrm c\\) are specified, to a first approximation, by the boundary conditions at the top and bottom layer surfaces. If the temperatures at these boundaries are fixed, the critical Rayleigh numbers are:

* for two stress-free boundaries

\\[\tag{3} R_c= \frac{27}{4}\pi ^4=657.511,\quad k_c=\frac{\pi}{\sqrt{2}}=2.221; \\]

* for two rigid boundaries

\\[\tag{4} R_c =1707.762,\quad k_c=3.117;\\]

* and for one rigid and one stress-free boundary

\\[\tag{5} R_c=1100.657,\quad k_c=2.682.\\]

Prior to Rayleigh's study, a necessary (but insufficient) condition of convective instability was found for compressible atmospheres (in the context of stability of the solar atmosphere) by K. Schwarzschild (1906) to be

\\[-\frac{\mathrm dT}{\mathrm dz}\equiv\left |\frac{\mathrm dT}{\mathrm dz}\right |>\left |\frac{\mathrm dT}{\mathrm dz}\right |_s,\\]

where \\(z\\) is the vertical coordinate (height) and the derivative on the right-hand side of the inequality refers to the isentropic stratification of the layer. This criterion came to be known as the Schwarzschild criterion. It has a clear physical meaning. Assume that the atmosphere is stratified adiabatically. If a fluid parcel that was initially in thermal and mechanical equilibrium with the ambient medium is displaced in a vertical direction from its initial position then, to a first approximation, its thermodynamic state experiences adiabatic changes and the parcel remains in equilibrium with the medium at any new height. If the temperature of the motionless medium varies with height more slowly and, accordingly, its density more rapidly than they would vary in the case of an adiabatic distribution, the parcel displaced upward proves to be heavier and the parcel displaced downward proves to be lighter than the medium. In both cases, the parcel will tend to return to its initial position. If, conversely, the temperature of the medium varies with height more rapidly and the density more slowly than in the adiabatic case, the density difference between the parcel and medium will produce a buoyancy force, which can result in the development of the convective instability. Dissipative factors, i.e., viscosity and thermal conduction, act so as to quench the convective instability. For this reason, the fact that the Schwarzschild criterion is satisfied is not sufficient for the development of convection. Since the driving force of convection in a compressible medium is directly related to the excess of the static temperature gradient (in its absolute magnitude) over the adiabatic gradient, precisely this excess appears, instead of the temperature gradient itself, in the equations of convection for a compressible medium.

A perfectly regular, spatially periodic flow described by the linear theory and observed in weakly supercritical conditions (at \\(R-R_\mathrm c \ll R_\mathrm c\\)) is constituted of identical "building blocks" (convection cells) closely packing the layer, and linear theory does not reveal any differences in the conditions of development of cells of various planforms. Most frequently, the following idealised planforms are considered:

* two-dimensional rolls (Fig. 2a);
* square cells (superpositions of two sets of mutually perpendicular rolls);
* hexagonal cells (superpositions of three sets of rolls rotated by an angle of \\(2\pi/3\\) to one another); hexagons of the so-called \\(l\\) and \\(g\\) types, shown in Fig. 2b, differ in the direction of circulation.

In experiments, nearly two-dimensional rolls (Fig. 3a) are typical of layers without any appreciable asymmetry of the physical conditions with respect to the horizontal midplane, under weakly or moderately supercritical conditions. If, however, such an up - down asymmetry is present (e.g., due to a temperature dependence of viscosity or other material properties of the fluid), three-dimensional cells arise (Fig. 3b).

## [edit] Supercritical regimes

As the Rayleigh number is increased, various flow instabilities can develop. The growth of numerous instability modes is possible, depending on the Rayleigh number, Prandtl number, horizontal wavenumber of the originally developed flow, and many other factors. As a result, the patterns of supercritical convection are diverse and highly sensitive not only to the parameters of the regime, \\(R\\) and \\(P\\), but also to the geometry of the container, to the boundary conditions, and to the prehistory of the flow. A comprehensive study of the instabilities of the convective flows was undertaken by F.H. Busse and his colleagues [see, in particular, Busse (1978, 1985); a fairly complete survey of these studies carried out by 1998 was given by Getling (1998)]. In particular, if the original flow forms a roll pattern, the rolls can acquire a wavelike form (the zig-zag instability), a secondary flow in the form of narrower rolls normal to the original ones can develop (the cross-roll instability), the rolls can undergo a deformation with asymmetrically located left-side and right-side broadenings (the skewed-varicose instability), the rolls can undulate (the oscillatory instability), etc. There are also particular instability modes that can transform the original roll flow into a three-dimensional flow that resembles a cell pattern.

At sufficiently high supercritical Rayleigh numbers, nonlinear effects can ultimately lead to the development of turbulence (Busse 1985; Koschmieder 1993; for a summary of some more recent results see Getling 1998).

## [edit] References

* H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Rev. Gén. Sciences Pure Appl., 11: 1261 - 1271, 11: 1309--1328, 1900.

* F.H. Busse, Non-linear properties of thermal convection, Rep. Prog. Phys., 41: 1929 - 1967, 1978.

* F.H. Busse, Transition to turbulence in Rayleigh - Bénard convection, In Hydrodynamic Instabilities and the Transition to Turbulence, eds. H.L. Swinney and J.P. Gollub (Topics in Appl. Phys., vol. 45), 2nd edition, Berlin, Springer, 1985, pp. 97 - 137.

* S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford, 1961; Dover Publications, Inc., New York, 1981.

* A. V. Getling, Rayleigh - Bénard Convection: Structures and Dynamics, World Scientific, Singapore, 1998 [Russian version: Editorial URSS, Moscow, 1999].

* E. L. Koschmieder, Bénard Cells and Taylor Vortices, Cambridge Univ. Press, Cambridge, 1993.

* A.C. Newell, T. Passot, and J. Lega, Order parameter equations for patterns, Ann. Rev. Fluid Mech., 25: 399 - 453, 1993.

* A. Pellew and R. V. Southwell, On maintained convective motion in a fluid heated from below, Proc. Roy. Soc. A, 176: 312 - 343, 1940.

* Lord Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag., Ser.6, 32: 529 - 546, 1916.

* F.M. Richter, Experiments on the stability of convection rolls in fluid whose viscosity depends on temperature, J. Fluid Mech., 89: 553 - 560, 1978.

* K. Schwarzschild, Über das Gleichgewicht der Sonnenatmosphäre, Nachr. Kgl. Ges. Wiss. Göttingen, Math.-Phys. Klasse, Nr. 1: 41 - 53, 1906.

## [edit] Further reading

* P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge Univ. Press, Cambridge, 1981.

* G.Z. Gershuni and E.M. Zhukovitskii, Convective Stability of Incompressible Fluids, Moscow, Nauka, 1972 [English translation: Jerusalem, Keter, 1976].

* D. D. Joseph, Stability of Fluid Motion: I and II (Springer tracts in natural philosophy, vols. 27 and 28), Berlin, Springer, 1976.