# Rayleigh number

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow[1][2][3], also known as free or natural convection. It is named after Lord Rayleigh.[4]. The Rayleigh number is used to describe fluids (such as water or air) when the mass density of the fluid is not uniform, but is higher in some parts of the fluid than in others. Gravity acts on these differences in mass density, to make the denser parts of the fluid fall, these falling of parts of the fluid is flow driven by gravity acting on mass density gradients, which is called convection.

When the Rayleigh number Ra is below a critical value for that fluid, there is no flow, whereas above it, the density difference in the fluid drives flow: convection[2]. Lord Rayleigh studied[1] the case of Rayleigh-Bénard convection[5]. Most commonly, the mass density differences are caused by temperature differences, typically fluids expand and become less dense as they are heated. Then, below the critical value of Ra. heat transfer is primarily in the form of diffusion of the thermal energy; when it exceeds the critical value, heat transfer is primarily in the form of convection.

When the mass density difference is caused by a temperature difference, Ra is, by definition, the ratio of the timescale for thermal transport due to thermal diffusion, to the timescale for thermal transport due to fluid falling at speed ${\displaystyle u}$under gravity[3]

${\displaystyle \mathrm {Ra} ={\frac {\mbox{timescale for thermal transport via diffusion}}{{\mbox{timescale for thermal transport via flow at speed}}~u}}}$

This means it is a type[3] of Péclet number. For a volume of fluid a size ${\displaystyle l}$across (in all three dimensions), with a mass density difference ${\displaystyle \Delta \rho }$, then the force of gravity is of order ${\displaystyle \Delta \rho l^{3}g}$, for ${\displaystyle g}$ the acceleration due to gravity. From the Stokes equation, when the volume of fluid is falling at speed ${\displaystyle u}$the viscous drag is of order ${\displaystyle \eta lu}$, for ${\displaystyle \eta }$ the viscosity of the fluid. Equating these forces we see that the speed ${\displaystyle u\sim \Delta \rho l^{2}g/\eta }$. So the timescale for transport via flow is ${\displaystyle l/u\sim \eta /\Delta \rho lg}$. The timescale for thermal diffusion across a distance ${\displaystyle l}$ is ${\displaystyle l^{2}/\alpha }$, where ${\displaystyle \alpha }$ is the thermal diffusivity. So the Rayleigh number Ra is

${\displaystyle \mathrm {Ra} ={\frac {l^{2}/\alpha }{\eta /\Delta \rho lg}}={\frac {\Delta \rho l^{3}g}{\eta \alpha }}={\frac {\rho \beta \Delta Tl^{3}g}{\eta \alpha }}}$

where we approximated the density difference ${\displaystyle \Delta \rho =\rho \beta \Delta T}$ for a fluid of average mass density ${\displaystyle \rho }$ with a thermal expansion coefficient ${\displaystyle \beta }$ and a temperature difference ${\displaystyle \Delta T}$ across the volume of fluid ${\displaystyle l}$across.

The Rayleigh number can be written as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity, ie Ra=Gr*Pr[3][2]. Hence it may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities.

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

${\displaystyle \mathrm {Ra} _{x}^{*}={\frac {g\beta q''_{o}}{\nu \alpha k}}x^{4}}$

where:

x is the characteristic length
Rax is the Rayleigh number for characteristic length x
q"o is the uniform surface heat flux
k is the thermal conductivity.[6]

For most engineering purposes, the Rayleigh number is large, somewhere around 106 to 108.

## Rayleigh-Darcy number for convection in a porous medium

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[7]. 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 equation, the falling speed of a domain of size ${\displaystyle l}$ of liquid ${\displaystyle u\sim \Delta \rho l^{2}g/\eta }$. In porous medium, this expression is replaced by that from Darcy's law ${\displaystyle u\sim \Delta \rho kg/\eta }$, with ${\displaystyle k}$ the permeability of the porous medium. The Rayleigh or Rayleigh-Darcy number is then

${\displaystyle \mathrm {Ra} ={\frac {\rho \beta \Delta Tklg}{\eta \alpha }}}$

This also applies to A-segregates, in the mushy zone of a solidifying alloy[8]. 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.[9] 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.[10]

### 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, RaH, is given by:

${\displaystyle \mathrm {Ra} _{H}={\frac {g\rho _{0}^{2}\beta HD^{5}}{\eta \alpha k}}}$

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.[11]

A Rayleigh number for bottom heating of the mantle from the core, RaT, can also be defined as:

${\displaystyle \mathrm {Ra} _{T}={\frac {\rho _{0}^{2}g\beta \Delta T_{sa}D^{3}C_{P}}{\eta k}}}$

where:

ΔTsa is the superadiabatic temperature difference between the reference mantle temperature and the core–mantle boundary
CP is the specific heat capacity at constant pressure.[11]

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.

## Notes

1. ^ a b Baron Rayleigh (1916). "On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side". London Edinburgh Dublin Phil. Mag. J. Sci. 32: 529–546.
2. ^ a b c Çengel, Yunus; Turner, Robert; Cimbala, John (2017). Fundamentals of thermal-fluid sciences (Fifth edition ed.). New York, NY. ISBN 9780078027680. OCLC 929985323.
3. ^ a b c d Squires, Todd M.; Quake, Stephen R. (2005-10-06). "Microfluidics: Fluid physics at the nanoliter scale". Reviews of Modern Physics. 77 (3): 977–1026. doi:10.1103/RevModPhys.77.977.
4. ^ Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. London: Oxford University Press. p. 10.
5. ^ Ahlers, Guenter; Grossmann, Siegfried; Lohse, Detlef (2009-04-22). "Heat transfer and large scale dynamics in turbulent Rayleigh-B\'enard convection". Reviews of Modern Physics. 81 (2): 503–537. arXiv:0811.0471. doi:10.1103/RevModPhys.81.503.
6. ^ M. Favre-Marinet and S. Tardu, Convective Heat Transfer, ISTE, Ltd, London, 2009
7. ^ Lister, John R.; Neufeld, Jerome A.; Hewitt, Duncan R. (2014). "High Rayleigh number convection in a three-dimensional porous medium". Journal of Fluid Mechanics. 748: 879–895. arXiv:0811.0471. doi:10.1017/jfm.2014.216. ISSN 1469-7645.
8. ^ Torabi Rad, M; Kotas, P; Beckermann, C (2013). "Rayleigh number criterion for formation of A-Segregates in steel castings and ingots". Metall. Mater. Trans. A. 44A: 4266–4281.
9. ^ Torabi Rad, M; Kotas, P; Beckermann, C (2013). "Rayleigh number criterion for formation of A-Segregates in steel castings and ingots". Metall. Mater. Trans. A. 44A: 4266–4281.
10. ^ Pickering, EJ; Al-Bermani, S; Talamantes-Silva, J (2014). "Application of criterion for A-segregation in steel ingots". Materials Science and Technology.
11. ^ a b Bunge, Hans-Peter; Richards, Mark A.; Baumgardner, John R. (1997). "A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number: Effects of depth-dependent viscosity, heating mode, and endothermic phase change". Journal of Geophysical Research. 102 (B6): 11991–12007. Bibcode:1997JGR...10211991B. doi:10.1029/96JB03806.

## References

• Turcotte, D.; Schubert, G. (2002). Geodynamics (2nd ed.). New York: Cambridge University Press. ISBN 0-521-66186-2.