In fluid thermodynamics, Rayleigh–Bénard convection is a type of

natural convection Convection is single or multiphase fluid flow that occurs spontaneously through the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the conve ...

, occurring in a planar horizontal layer of

fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...

heated from below, in which the fluid develops a regular pattern of

convection cells In fluid dynamics, a convection cell is the phenomenon that occurs when density differences exist within a body of liquid or gas. These density differences result in rising and/or falling convection currents, which are the key characteristics ...

known as Bénard cells. Such systems were first investigated by

Joseph Valentin Boussinesq Joseph Valentin Boussinesq (; 13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat. Biography From 1872 to 1886, he was appoin ...

and Anton Oberbeck in the 19th century. This phenomenon can also manifest where a species denser than the electrolyte is consumed from below and generated at the top. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing

nonlinear system In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathem ...

s. Time-dependent

self-similar In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar ...

analytic solutions are known for the velocity fields and for the temperature distribution as well.

Buoyancy Buoyancy (), or upthrust, is the force exerted by a fluid opposing the weight of a partially or fully immersed object (which may be also be a parcel of fluid). In a column of fluid, pressure increases with depth as a result of the weight of t ...

, and hence

gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

, are responsible for the appearance of convection cells. The initial movement is the

upwelling Upwelling is an physical oceanography, oceanographic phenomenon that involves wind-driven motion of dense, cooler, and usually nutrient-rich water from deep water towards the ocean surface. It replaces the warmer and usually nutrient-depleted sur ...

of less-

dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...

fluid from the warmer bottom layer. This upwelling spontaneously organizes into a regular pattern of cells. Tiger striping grass

Physical processes

The features of Bénard convection can be obtained by a simple experiment first conducted by Henri Bénard, a French physicist, in 1900.

Development of convection

The experimental set-up uses a layer of liquid, e.g. water, between two parallel planes. The height of the layer is small compared to the horizontal dimension. At first, the temperature of the bottom plane is the same as the top plane. The liquid will then tend towards an

equilibrium Equilibrium may refer to: Film and television * ''Equilibrium'' (film), a 2002 science fiction film * '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film * "Equilibrium" (''seaQuest 2032'') * ''Equilibr ...

, where its temperature is the same as its surroundings. (Once there, the liquid is perfectly uniform: to an observer it would appear the same from any position. This equilibrium is also

asymptotically stable Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...

: after a local, temporary perturbation of the outside temperature, it will go back to its uniform state, in line with the

second law of thermodynamics The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...

). Then, the temperature of the bottom plane is increased slightly yielding a flow of thermal energy conducted through the liquid. The system will begin to have a structure of

thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...

: the temperature, and the density and pressure with it, will vary linearly between the bottom and top plane. A uniform linear gradient of temperature will be established. (This system may be modelled by

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

). Once conduction is established, the microscopic random movement ''spontaneously'' becomes ordered on a macroscopic level, forming Benard convection cells, with a characteristic correlation length.

Convection features

The rotation of the cells is stable and will alternate from clock-wise to counter-clockwise horizontally; this is an example of

spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion o ...

. Bénard cells are

metastable In chemistry and physics, metastability is an intermediate energetic state within a dynamical system other than the system's state of least energy. A ball resting in a hollow on a slope is a simple example of metastability. If the ball is onl ...

. This means that a small perturbation will not be able to change the rotation of the cells, but a larger one could affect the rotation; they exhibit a form of

hysteresis Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...

. Moreover, the deterministic law at the microscopic level produces a non-deterministic arrangement of the cells: if the experiment is repeated, a particular position in the experiment will be in a clockwise cell in some cases, and a counter-clockwise cell in others. Microscopic perturbations of the

initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). Fo ...

are enough to produce a non-deterministic macroscopic effect. That is, in principle, there is no way to calculate the macroscopic effect of a microscopic perturbation. This inability to predict long-range conditions and sensitivity to initial-conditions are characteristics of chaotic or

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

systems (i.e., the

butterfly effect In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The term is closely associated w ...

). If the temperature of the bottom plane were to be further increased, the structure would become more complex in space and time; the

turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by Chaos theory, chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disrupt ...

would become chaotic. Convective Bénard cells tend to approximate regular right hexagonal prisms, particularly in the absence of turbulence, although certain experimental conditions can result in the formation of regular right square prisms or spirals. The convective Bénard cells are not unique and will usually appear only in the surface tension driven convection. In general the solutions to the Rayleigh and Pearson analysis (linear theory) assuming an infinite horizontal layer gives rise to degeneracy meaning that many patterns may be obtained by the system. Assuming uniform temperature at the top and bottom plates, when a realistic system is used (a layer with horizontal boundaries) the shape of the boundaries will mandate the pattern. More often than not the convection will appear as rolls or a superposition of them.

Rayleigh–Bénard instability

Since there is a density gradient between the top and the bottom plate, gravity acts trying to pull the cooler, denser liquid from the top to the bottom. This gravitational force is opposed by the viscous damping force in the fluid. The balance of these two forces is expressed by a non-dimensional parameter called the

Rayleigh number In fluid mechanics, the Rayleigh number (, after Lord Rayleigh) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. It characterises the fluid's flow regime: a value in a certa ...

. The Rayleigh number is defined as: :

\mathrm_ = \frac  (T_b - T_u) L^3

where :''T_u'' is the temperature of the top plate :''T_b'' is the temperature of the bottom plate :''L'' is the height of the container :''g'' is the acceleration due to gravity :''ν'' is the

kinematic viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

:''α'' is the

thermal diffusivity In thermodynamics, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It is a measure of the rate of heat transfer inside a material and has SI, SI units of m2/s. It is an intensive ...

:''β'' is the

thermal expansion coefficient Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions). Substances usually contract with decreasing temp ...

. As the Rayleigh number increases, the gravitational forces become more dominant. At a critical Rayleigh number of 1708, instability sets in and convection cells appear. The critical Rayleigh number can be obtained analytically for a number of different boundary conditions by doing a perturbation analysis on the linearized equations in the stable state. The simplest case is that of two free boundaries, which

Lord Rayleigh John William Strutt, 3rd Baron Rayleigh ( ; 12 November 1842 – 30 June 1919), was an English physicist who received the Nobel Prize in Physics in 1904 "for his investigations of the densities of the most important gases and for his discovery ...

solved in 1916, obtaining Ra = π⁴ ≈ 657.51. In the case of a rigid boundary at the bottom and a free boundary at the top (as in the case of a kettle without a lid), the critical Rayleigh number comes out as Ra = 1,100.65.

Effects of surface tension

In case of a free liquid surface in contact with air, buoyancy and

surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...

effects will also play a role in how the convection patterns develop. Liquids flow from places of lower surface tension to places of higher surface tension. This is called the

Marangoni effect The Marangoni effect (also called the Gibbs–Marangoni effect) is the mass transfer along an Interface (chemistry), interface between two phases due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may ...

. When applying heat from below, the temperature at the top layer will show temperature fluctuations. With increasing temperature, surface tension decreases. Thus a lateral flow of liquid at the surface will take place, from warmer areas to cooler areas. In order to preserve a horizontal (or nearly horizontal) liquid surface, cooler surface liquid will descend. This down-welling of cooler liquid contributes to the driving force of the convection cells. The specific case of temperature gradient-driven surface tension variations is known as thermo-capillary convection, or Bénard–Marangoni convection.

History and nomenclature

In 1870, the Irish-Scottish physicist and engineer James Thomson (1822–1892), elder brother of

Lord Kelvin William Thomson, 1st Baron Kelvin (26 June 182417 December 1907), was a British mathematician, Mathematical physics, mathematical physicist and engineer. Born in Belfast, he was the Professor of Natural Philosophy (Glasgow), professor of Natur ...

, observed water cooling in a tub; he noted that the soapy film on the water's surface was divided as if the surface had been tiled (tesselated). In 1882, he showed that the tesselation was due to the presence of convection cells. In 1900, the French physicist Henri Bénard independently arrived at the same conclusion. This pattern of convection, whose effects are due solely to a temperature gradient, was first successfully analyzed in 1916 by

. Rayleigh assumed boundary conditions in which the vertical velocity component and temperature disturbance vanish at the top and bottom boundaries (perfect thermal conduction). Those assumptions resulted in the analysis losing any connection with Henri Bénard's experiment. This resulted in discrepancies between theoretical and experimental results until 1958, when John Pearson (1930– ) reworked the problem based on surface tension. This is what was originally observed by Bénard. Nonetheless in modern usage "Rayleigh–Bénard convection" refers to the effects due to temperature, whereas "Bénard–Marangoni convection" refers specifically to the effects of surface tension. Davis and Koschmieder have suggested that the convection should be rightfully called the "Pearson–Bénard convection". Rayleigh–Bénard convection is also sometimes known as "Bénard–Rayleigh convection", "Bénard convection", or "Rayleigh convection".

References

External links

A. Getling, O. Brausch: Cellular flow patternsK. Daniels, B. Plapp, W.Pesch, O. Brausch, E. Bodenschatz: Undulation Chaos in inclined Layer ConvectionKaren E. Daniels, Oliver Brausch, Werner Pesch, Eberhard Bodenschatz: Competition and bistability of ordered undulations and undulation chaos in inclined layer convection
(PDF; 608 kB)
P. Subramanian, O. Brausch, E. Bodenschatz, K. Daniels, T.Schneider W. Pesch: Spatio-temporal Patterns in Inclined Layer Convection
(PDF; 5,3 MB) {{DEFAULTSORT:Benard Cells Convection Fluid dynamic instabilities Articles containing video clips