Editing Rayleigh number

{{short description|Dimensionless quantity associated with free convection of a fluid}}

In [[fluid mechanics]], the '''Rayleigh number''' ({{math|'''Ra'''}}, after [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]]<ref>{{cite book |last1=Chandrasekhar |first1=S. |title=Hydrodynamic and Hydromagnetic Stability |url=https://archive.org/details/hydrodynamichydr00chan_094 |url-access=limited |date=1961 |publisher=Oxford University Press |location=London |page=[https://archive.org/details/hydrodynamichydr00chan_094/page/n15 10]|isbn=978-0-19-851237-0 }}</ref>) for a [[fluid]] is a [[dimensionless number]] associated with [[buoyancy]]-driven flow, also known as [[free convection|free]] (or natural) [[convection]].<ref name=":0">{{Cite journal |last=Baron Rayleigh |date=1916 |title=On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side |journal=London Edinburgh Dublin Phil. Mag. J. Sci. |volume=32 |issue=192 |pages=529–546 |doi=10.1080/14786441608635602|url=https://zenodo.org/record/1508106 }}</ref><ref name=":2">{{Cite book |title=Fundamentals of thermal-fluid sciences |last1=Çengel |first1=Yunus |last2=Turner |first2=Robert |last3=Cimbala |first3=John |year=2017 |isbn=9780078027680 |edition= Fifth |location=New York, NY |oclc=929985323}}</ref><ref name=":1"/> It characterises the fluid's flow regime:<ref name=":3-p466">{{cite book |last1=Çengel |first1=Yunus A. |title=Heat and Mass Transfer |date=2002 |publisher=McGraw-Hill |page=466 |edition=Second}}</ref> a value in a certain lower range denotes [[laminar flow]]; a value in a higher range, [[turbulence|turbulent flow]]. Below a certain critical value, there is no fluid motion and [[heat transfer]] is by [[thermal conduction|conduction]] rather than convection. For most engineering purposes, the Rayleigh number is large, somewhere around 10<sup>6</sup> to 10<sup>8</sup>.

The Rayleigh number is defined as the product of the [[Grashof number]] ({{math|Gr}}), which describes the relationship between [[buoyancy]] and [[viscosity]] within a fluid, and the [[Prandtl number]] ({{math|Pr}}), which describes the relationship between [[kinematic viscosity|momentum diffusivity]] and [[thermal diffusivity]]: {{math|1=Ra = Gr × Pr}}.<ref name=":1"/><ref name=":2"/> Hence it may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities: {{math|1=Ra = B/''μ'' × ''ν''/''α''}}. It is closely related to the [[Nusselt number]] ({{math|Nu}}).<ref name=":3-p466"/>

==Derivation==
The Rayleigh number describes the behaviour of fluids (such as water or air) when the mass density of the fluid is non-uniform. The mass density differences are usually caused by temperature differences. Typically a fluid expands and becomes less dense as it is heated. Gravity causes denser parts of the fluid to sink, which is called [[convection]]. Lord Rayleigh studied<ref name=":0"/> the case of [[Rayleigh–Bénard convection|Rayleigh-Bénard convection]].<ref>{{Cite journal |last1=Ahlers |first1=Guenter |last2=Grossmann| first2=Siegfried |last3=Lohse |first3=Detlef |date=2009-04-22 |title=Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection |journal=Reviews of Modern Physics |volume=81 |issue=2 |pages=503–537 |arxiv=0811.0471 |doi=10.1103/RevModPhys.81.503|bibcode=2009RvMP...81..503A |s2cid=7566961 }}</ref> When the Rayleigh number, Ra, is below a critical value for a fluid, there is no flow and heat transfer is purely by [[thermal conduction|conduction]]; when it exceeds that value, heat is transferred by natural convection.<ref name=":2"/>

When the mass density difference is caused by temperature difference, Ra is, by definition, the ratio of the time scale for diffusive thermal transport to the time scale for convective thermal transport at speed <math>u</math>:<ref name=":1">{{Cite journal |last1=Squires |first1=Todd M. |last2=Quake |first2=Stephen R. |date=2005-10-06 |title=Microfluidics: Fluid physics at the nanoliter scale |journal=Reviews of Modern Physics |volume=77 |issue=3 |pages=977–1026 |doi=10.1103/RevModPhys.77.977 |url=https://authors.library.caltech.edu/1310/1/SQUrmp05.pdf |bibcode=2005RvMP...77..977S}}</ref>

<math display="block">\mathrm{Ra} = \frac{\text{time scale for thermal transport via diffusion}}{\text{time scale for thermal transport via convection at speed}~ u}.</math>

This means the Rayleigh number is a type<ref name=":1"/> of [[Péclet number]]. For a volume of fluid of size <math>l</math> in all three dimensions{{clarify|reason=e.g. Does this mean a cube of side-length l?|date=December 2020}} and mass density difference <math>\Delta\rho</math>, the force due to gravity is of the order <math>\Delta\rho l^3g</math>, where <math>g</math> is acceleration due to gravity. From the [[Stokes's law|Stokes equation]], when the volume of fluid is sinking, viscous drag is of the order <math>\eta l u</math>, where <math>\eta</math> is the [[dynamic viscosity]] of the fluid. When these two forces are equated, the speed <math>u \sim \Delta\rho l^2 g/\eta</math>. Thus the time scale for transport via flow is <math>l/u \sim \eta/\Delta\rho lg</math>. The time scale for thermal diffusion across a distance <math>l</math> is <math>l^2/\alpha</math>, where <math>\alpha</math> is the [[thermal diffusivity]]. Thus the Rayleigh number Ra is

<math display="block">\mathrm{Ra} = \frac{l^2/\alpha}{\eta/\Delta\rho lg} = \frac{\Delta\rho l^3g}{\eta\alpha} = \frac{\rho\beta\Delta T l^3g}{\eta\alpha}</math>

where we approximated the density difference <math>\Delta\rho=\rho\beta\Delta T</math> for a fluid of average mass density <math>\rho</math>, [[Coefficient of thermal expansion|thermal expansion coefficient]] <math>\beta</math> and a temperature difference <math>\Delta T</math> across distance <math>l</math>.

The Rayleigh number can be written as the product of the [[Grashof number]] and the [[Prandtl number]]:<ref name=":1"/><ref name=":2"/>
<math display="block">\mathrm{Ra} = \mathrm{Gr}\mathrm{Pr}.</math>

==Classical definition==
For [[free convection]] near a vertical wall, the Rayleigh number is defined as:

<math display="block">\mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infty) x^3 = \mathrm{Gr}_{x}\mathrm{Pr}</math>

where:
*''x'' is the characteristic length
*Ra<sub>''x''</sub> is the Rayleigh number for characteristic length ''x''
*''g'' is acceleration due to gravity
*''β'' is the [[Coefficient of thermal expansion|thermal expansion coefficient]] (equals to 1/''T'', for ideal gases, where ''T'' is absolute temperature).
*<math>\nu</math> is the [[kinematic viscosity]]
*''α'' is the [[thermal diffusivity]]
*''T<sub>s</sub>'' is the surface temperature
*''T''<sub>∞</sub> is the quiescent temperature (fluid temperature far from the surface of the object)
*Gr<sub>''x''</sub> is the [[Grashof number]] for characteristic length ''x''
*Pr is the [[Prandtl number]]

In the above, the fluid properties Pr, ''ν'', ''α'' and ''β'' are evaluated at the [[film temperature]], which is defined as:

<math display="block">T_f = \frac{T_s + T_\infin}{2}.</math>

For a uniform wall heating flux, the modified Rayleigh number is defined as:

<math display="block">\mathrm{Ra}^{*}_{x} = \frac{g \beta q''_o} {\nu \alpha k} x^4 </math>

where:
*''q&Prime;<sub>o</sub>'' is the uniform surface heat flux
*''k'' is the thermal conductivity.<ref>M. Favre-Marinet and S. Tardu, Convective Heat Transfer, ISTE, Ltd, London, 2009</ref>

==Other applications==
===Solidifying alloys===
The Rayleigh number can also be used as a criterion to predict convectional instabilities, such as [[A-segregates]], in the mushy zone of a solidifying alloy. The mushy zone Rayleigh number is defined as:

<math display="block">\mathrm{Ra} = \frac{\frac{\Delta \rho}{\rho_0}g \bar{K} L}{\alpha \nu} = \frac{\frac{\Delta \rho}{\rho_0}g \bar{K} }{R \nu}</math>

where:
*''<span style="text-decoration: overline">K</span>'' is the mean permeability (of the initial portion of the mush)
*''L'' is the characteristic length scale 
*''α'' is the thermal diffusivity 
*''ν'' is the kinematic viscosity 
*''R'' is the solidification or isotherm speed.<ref name="ReferenceA">{{cite journal |last1=Torabi Rad |first1=M. |last2=Kotas |first2=P. |last3=Beckermann |first3=C. |author3-link=Christoph Beckermann |title=Rayleigh number criterion for formation of A-Segregates in steel castings and ingots |journal=Metall. Mater. Trans. A |date=2013 |volume=44A |issue=9 |pages=4266–4281|doi=10.1007/s11661-013-1761-4 |bibcode=2013MMTA...44.4266R |s2cid=137652216 }}</ref>

A-segregates are predicted to form when the Rayleigh number exceeds a certain critical value. This critical value is independent of the composition of the alloy, and this is the main advantage of the Rayleigh number criterion over other criteria for prediction of convectional instabilities, such as Suzuki criterion.

Torabi Rad et al. showed that for steel alloys the critical Rayleigh number is 17.<ref name="ReferenceA"/> Pickering et al. explored Torabi Rad's criterion, and further verified its effectiveness. Critical Rayleigh numbers for lead–tin and nickel-based super-alloys were also developed.<ref>{{cite journal |last1=Pickering |first1=E.J. |last2=Al-Bermani |first2=S. |last3=Talamantes-Silva |first3=J. |title=Application of criterion for A-segregation in steel ingots |journal=Materials Science and Technology |date=2014|volume=31 |issue=11 |page=1313 |doi=10.1179/1743284714Y.0000000692 |bibcode=2015MatST..31.1313P |s2cid=137549220 }}</ref>

===Porous media===
The Rayleigh number above is for convection in a bulk fluid such as air or water, but convection can also occur when the fluid is inside and fills a porous medium, such as porous rock saturated with water.<ref>{{Cite journal |last1=Lister |first1=John R. |last2=Neufeld |first2=Jerome A. |last3=Hewitt |first3=Duncan R. |date=2014 |title=High Rayleigh number convection in a three-dimensional porous medium |journal=Journal of Fluid Mechanics|language=en|volume=748|pages=879–895 |arxiv=0811.0471 |doi=10.1017/jfm.2014.216 |bibcode=2014JFM...748..879H |s2cid=43758157 |issn=1469-7645}}</ref> Then the Rayleigh number, sometimes called the '''Rayleigh-Darcy number''', is different. In a bulk fluid, i.e., not in a porous medium, from the [[Stokes's law|Stokes equation]], the falling speed of a domain of size <math>l</math> of liquid <math>u \sim \Delta\rho l^2 g/\eta</math>. In porous medium, this expression is replaced by that from [[Darcy's law]] <math>u \sim \Delta\rho k g/\eta</math>, with <math>k</math> the permeability of the porous medium. The Rayleigh or Rayleigh-Darcy number is then

<math display="block">\mathrm{Ra}=\frac{\rho\beta\Delta T klg}{\eta\alpha}</math>

This also applies to [[A-segregates]], in the mushy zone of a solidifying alloy.<ref name="ReferenceA"/>

===Geophysical applications===
In [[geophysics]], the Rayleigh number is of fundamental importance: it indicates the presence and strength of convection within a fluid body such as the [[Earth's mantle]]. The mantle is a solid that behaves as a fluid over geological time scales. The Rayleigh number for the Earth's mantle due to internal heating alone, Ra''<sub>H</sub>'', is given by:

<math display="block">\mathrm{Ra}_H = \frac{g\rho^{2}_{0}\beta HD^5}{\eta \alpha k}</math>

where:
*''H'' is the rate of [[radiogenic heat]] production per unit mass
*''η'' is the [[dynamic viscosity]]
*''k'' is the [[thermal conductivity]]
*''D'' is the depth of the [[Mantle (geology)|mantle]].<ref name=Bunge>{{cite journal |first1=Hans-Peter |last1=Bunge |last2=Richards |first2=Mark A. |last3=Baumgardner |first3=John R. |year=1997 |title=A sensitivity study of three-dimensional spherical mantle convection at 10<sup>8</sup> Rayleigh number: Effects of depth-dependent viscosity, heating mode, and endothermic phase change| journal = [[Journal of Geophysical Research]] | pages = 11991–12007 | volume = 102 |issue=B6 |doi=10.1029/96JB03806 |bibcode = 1997JGR...10211991B |doi-access=free }}</ref>

A Rayleigh number for bottom heating of the mantle from the core, Ra''<sub>T</sub>'', can also be defined as:

<math display="block">\mathrm{Ra}_T = \frac{\rho_{0}^2 g\beta\Delta T_\text{sa}D^3 C_P}{\eta k}</math>

where:
*Δ''T''<sub>sa</sub> is the superadiabatic temperature difference (the superadiabatic temperature difference is the actual temperature difference minus the temperature difference in a fluid whose [[entropy]] gradient is zero, but has the same profile of the other variables appearing in the [[equation of state]]) between the reference mantle temperature and the [[core–mantle boundary]]
*''C<sub>P</sub>'' is the [[specific heat capacity]] at constant pressure.<ref name=Bunge />

High values for the Earth's mantle indicates that convection within the Earth is vigorous and time-varying, and that convection is responsible for almost all the heat transported from the deep interior to the surface.

==See also==
* [[Grashof number]]
* [[Prandtl number]]
* [[Reynolds number]]
* [[Péclet number]]
* [[Nusselt number]]
* [[Rayleigh–Bénard convection]]

==Notes==
{{Reflist}}

== References ==
*{{cite book |title=Geodynamics |first1=D. |last1=Turcotte |first2=G. |last2=Schubert |edition= 2nd |year=2002 |location=New York |publisher=Cambridge University Press |isbn=978-0-521-66186-7 }}

==External links==
* [https://www.fxsolver.com/browse/formulas/Rayleigh+Number Rayleigh number calculator]

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