# 瑞利-贝纳德对流

Bénard cells.

Bénard cells.

Rayleigh-Bénard convection is a type of natural convection, occurring in a plane horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility.[1] The convection patterns are the most carefully examined example of self-organizing nonlinear systems.[1][2]

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. Bénard cells are metastable. 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.

Buoyancy, and hence gravity, are responsible for the appearance of convection cells. The initial movement is the upwelling of lesser density fluid from the heated bottom layer.[3] This upwelling spontaneously organizes into a regular pattern of cells.

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.

## Physical processes

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

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. The Rayleigh number is defined as:

### Development of convection

$\displaystyle{ \mathrm{Ra}_{L} = \frac{g \beta} {\nu \alpha} (T_b - T_u) L^3 }$

(t _ b-t _ u) l ^ 3 </math >

Convection cells in a gravity field

where

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, 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: 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).

Tu is the temperature of the top plate

T < sub > u 是顶板的温度

Tb is the temperature of the bottom plate

T < sub > b 是底板的温度

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 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).

L is the height of the container

L 是容器的高度

g is the acceleration due to gravity

G 是由重力引起的加速度

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

ν is the kinematic viscosity

ν 是运动粘度

α is the Thermal diffusivity

α 是热扩散率

### Convection features

β is the Thermal expansion coefficient.

β 是热膨胀系数。

Simulation of Rayleigh–Bénard convection in 3D.

As the Rayleigh number increases, the gravitational forces become more dominant. At a critical Rayleigh number of 1708, The simplest case is that of two free boundaries, which Lord Rayleigh 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.

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. Bénard cells are metastable. 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.

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 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 systems (i.e., the butterfly effect).

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

turbulent Rayleigh–Bénard convection

If the temperature of the bottom plane was to be further increased, the structure would become more complex in space and time; the turbulent flow would become chaotic.

In 1870, the Irish-Scottish physicist and engineer James Thomson (1822–1892), elder brother of Lord Kelvin, 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 (1874–1939) 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 Lord Rayleigh (1842–1919). 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".

1870年，爱尔兰-苏格兰物理学家和工程师詹姆斯 · 汤姆森(1822-1892) ，开尔文勋爵的哥哥，观察到水在浴缸中冷却，他注意到水面上的肥皂膜被分开，好像表面被瓷砖(镶嵌)。1882年，他证明了镶嵌是由于对流环的存在。1900年，法国物理学家亨利 · 贝纳德(1874-1939)独立地得出了同样的结论。这种对流的模式，其影响完全是由于温度梯度，首次成功地分析在1916年的主瑞利(1842-1919年)。瑞利假设的边界条件中，垂直速度分量和温度扰动在顶部和底部边界消失(完全热传导)。这些假设导致分析失去了与亨利 · 贝纳德实验的任何联系。这导致了理论和实验结果之间的差异，直到1958年，当约翰皮尔森(1930 -)重新工作的基础上表面张力的问题。这是贝纳德最初观察到的现象。尽管如此，在现代用法中，“ Rayleigh-Bénard 对流”是指温度的影响，而“ Bénard-Marangoni 对流”是指表面张力的影响。戴维斯和科施米德建议，对流应正确地称为“皮尔逊-贝纳德对流”。

Convective Bénard cells tend to approximate regular right hexagonal prisms, particularly in the absence of turbulence,[4][5][6] although certain experimental conditions can result in the formation of regular right square prisms[7] or spirals.[8]

Rayleigh–Bénard convection is also sometimes known as "Bénard–Rayleigh convection", "Bénard convection", or "Rayleigh convection".

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[9] 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. The Rayleigh number is defined as:

$\displaystyle{ \mathrm{Ra}_{L} = \frac{g \beta} {\nu \alpha} (T_b - T_u) L^3 }$

where

Tu is the temperature of the top plate
Tb 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
α is the Thermal diffusivity
β is the Thermal expansion coefficient.

As the Rayleigh number increases, the gravitational forces become more dominant. At a critical Rayleigh number of 1708,[2] 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.[10] The simplest case is that of two free boundaries, which Lord Rayleigh solved in 1916, obtaining Ra = 模板:Frac π4 ≈ 657.51.[11] 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.[12]

## Effects of surface tension

In case of a free liquid surface in contact with air, buoyancy and surface tension 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. 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,[13] 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, 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.[14] In 1900, the French physicist Henri Bénard (1874–1939) independently arrived at the same conclusion.[15] This pattern of convection, whose effects are due solely to a temperature gradient, was first successfully analyzed in 1916 by Lord Rayleigh (1842–1919).[16] 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.[9] 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.[1] Davis and Koschmieder have suggested that the convection should be rightfully called the "Pearson–Bénard convection".[2]

Rayleigh–Bénard convection is also sometimes known as "Bénard–Rayleigh convection", "Bénard convection", or "Rayleigh convection".

Category:Convection

Category:Fluid dynamic instability

Category:Articles containing video clips

This page was moved from wikipedia:en:Rayleigh–Bénard convection. Its edit history can be viewed at 瑞利-贝纳德对流/edithistory

1. Rayleigh-Bénard convection is a type of natural convection, occurring in a plane horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. 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 systems. 瑞利-贝纳德对流是自然对流的一种，它发生在一个从下面加热的平面水平流体层中，在这个流体层中形成一种称为贝纳德细胞的有规律的对流格局。贝纳德-瑞利对流是研究最多的对流现象之一，因为它具有解析和实验的可及性。对流模式是自组织非线性系统的最仔细的研究例子。 Getling, A. V. Buoyancy, and hence gravity, are responsible for the appearance of convection cells. The initial movement is the upwelling of lesser density fluid from the heated bottom layer. This upwelling spontaneously organizes into a regular pattern of cells. 浮力，也就是重力，负责对流单元的出现。最初的运动是较小密度的流体从加热的底层上涌。这种上涌自发地组织成一种有规律的细胞模式。 (1998). Bénard–Rayleigh Convection: Structures and Dynamics. World Scientific The features of Bénard convection can be obtained by a simple experiment first conducted by Henri Bénard, a French physicist, in 1900. 1900年，法国物理学家亨利 · 贝纳德首先进行了一个简单的实验，得到了贝纳德对流的特征。这个实验阐明了耗散系统的理论。< = = 这句话需要在上下文中解释 -- >. ISBN 978-981-02-2657-2.
2. {{cite book Convection cells in a gravity field 重力场中的对流单元 |last=Koschmieder 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, 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: 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). 实验装置使用一层液体，例如:。水，在两个平行平面之间。与水平方向相比，地层的高度较小。首先，底面的温度与顶面的温度相同。然后液体趋于平衡，其温度与周围环境相同。(一旦到达那里，液体是完全均匀的: 对于观察者来说，从任何位置看都是一样的。这个平衡也是渐近稳定的: 在一个局部的，暂时的外部温度扰动之后，它将回到它的均匀状态，与热力学第二定律一致)。 |first=E. L. |year=1993 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 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). 然后，底面的温度略有增加，产生一股热能流通过液体。该系统将开始有一个热导率的结构: 温度，密度和压力，将与它的底部和顶部平面之间线性变化。将建立一个均匀的线性温度梯度。(这个系统可能是以统计力学为模型的)。 |title=Bénard Cells and Taylor Vortices |publisher=Cambridge Once conduction is established, the microscopic random movement spontaneously becomes ordered on a macroscopic level, forming Benard convection cells, with a characteristic correlation length. 一旦导热建立，微观随机运动在宏观水平上自发地变得有序，形成具有特征相关长度的贝纳德对流单元。 |isbn=0521-40204-2 |url-access=registration |url=https://archive.org/details/benardcellstaylo0000kosc Simulation of Rayleigh–Bénard convection in 3D. 三维瑞利-贝纳德对流的数值模拟。 }}
3. 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 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 systems (i.e., the butterfly effect). 此外，微观层面的确定性规律产生了细胞的非确定性排列: 如果实验重复进行，在某些情况下，实验中的特定位置在顺时针的细胞中，在另一些情况下则在逆时针的细胞中。初始条件的微观扰动足以产生不确定的宏观效应。也就是说，原则上，没有办法计算微观扰动的宏观效应。这种无法预测长期条件和对初始条件的敏感性是混沌或复杂系统的特征(即蝴蝶效应)。 {{cite web < ! ——也许还表现出自组织临界性或作为一个动力系统? —— > |title=Rayleigh–Benard Convection turbulent Rayleigh–Bénard convection 湍流瑞利-贝纳德对流 |url=http://physics.ucsd.edu/was-daedalus/convection/rb.html If the temperature of the bottom plane was to be further increased, the structure would become more complex in space and time; the turbulent flow would become chaotic. 如果底面温度进一步升高，结构在时间和空间上都将变得更加复杂，湍流将变得混沌。 |publisher=UC San Diego, Department of Physics |archiveurl=https://web.archive.org/web/20090222182327/http://physics.ucsd.edu/was-daedalus/convection/rb.html 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. 对流的 b énard 单元倾向于近似于规则的右六角棱镜，特别是在没有湍流的情况下，虽然某些实验条件可能导致规则的右方棱镜或螺旋的形成。 |archivedate=22 February 2009 }}
4. Rayleigh–Benard Convection Cells, with photos, from the Environmental Technology Laboratory at the National Oceanic and Atmospheric Administration in the United States Department of Commerce.
5. http://www.edata-center.com/proceedings/1bb331655c289a0a,088ce8ea747789cd,59d115f133a4fd07.html
6. http://cat.inist.fr/?aModele=afficheN&cpsidt=17287579
7. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=13973
8. http://www.psc.edu/science/Gunton/gunton.html
9. Pearson, J.R.A. (1958). "On convection cells induced by surface tension". Journal of Fluid Mechanics. 4 (5): 489–500. doi:10.1017/S0022112058000616.
10. http://home.iitk.ac.in/~sghorai/NOTES/benard/benard.html
11. http://home.iitk.ac.in/~sghorai/NOTES/benard/node14.html
12. http://home.iitk.ac.in/~sghorai/NOTES/benard/node16.html
13. Steady thermocapillary flows in two-dimensional slots Journal of Fluid Mechanics, Vol. 121 (1982), pp. 163-186, doi:10.1017/s0022112082001840 by Asok K. Sen, Stephen H. Davis
14. Thomson, James (1882). "On a changing tesselated structure in certain liquids". Proceedings of the Philosophical Society of Glasgow. 8 (2): 464–468.
15. Bénard, Henri (1900). "Les tourbillons cellulaires dans une nappe liquide" [Cellular vortices in a sheet of liquid]. Revue Générale des Sciences Pures et Appliquées (in French). 11: 1261–1271, 1309–1328.{{cite journal}}: CS1 maint: unrecognized language (link)
16. Rayleigh, Lord (1916). "On the convective currents in a horizontal layer of fluid when the higher temperature is on the under side". Philosophical Magazine. 6th series. 32 (192): 529–546.