In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy driven flow (also known as free convection or natural convection). When the Rayleigh number is below the critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.
The Rayleigh number is named after Lord Rayleigh and is defined 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. Hence the Rayleigh number itself may also be viewed as the ratio of buoyancy and viscosity forces times the ratio of momentum and thermal diffusivities.
Definition
For free convection near a vertical wall, the Rayleigh number is defined as

\mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s  T_\infin) x^3 = \mathrm{Gr}_{x}\mathrm{Pr}
where

x = Characteristic length (in this case, the distance from the leading edge)

Ra_{x} = Rayleigh number at position x

Gr_{x} = Grashof number at position x

Pr = Prandtl number

g = acceleration due to gravity

T_{s} = Surface temperature (temperature of the wall)

T_{∞} = Quiescent temperature (fluid temperature far from the surface of the object)

ν = Kinematic viscosity

α = Thermal diffusivity

β = Thermal expansion coefficient (equals to 1/T, for ideal gases, where T is absolute temperature)
In the above, the fluid properties Pr, ν, α and β are evaluated at the film temperature, which is defined as
T_f = \frac{T_s + T_\infin}{2}
For most engineering purposes, the Rayleigh number is large, somewhere around 10^{6} to 10^{8}.
For a uniform wall heating flux, the modified Rayleigh number is defined as ^{[1]}
\mathrm{Ra}^{*}_{x} = \frac{g \beta q''_o} {\nu \alpha k} x^4
where

q"_{o} = the uniform surface heat flux (W/m^{2})

k = the thermal conductivity (W/m•K)
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_{H} is given by
\mathrm{Ra}_H = \frac{g\rho^{2}_{0}\beta HD^5}{\eta \alpha k}
where H is the rate of radiogenic heat production, η is the dynamic viscosity, k is the thermal conductivity, and D is the depth of the mantle.^{[2]}
A Rayleigh number for bottom heating of the mantle from the core, Ra_{T} can also be defined:
\mathrm{Ra}_T = \frac{\rho_{0}g\beta\Delta T_{sa}D^3 c}{\eta k} ^{[2]} Where ΔT_{sa} is the superadiabatic temperature difference between the reference mantle temperature and the Core–mantle boundary and c is the specific heat capacity, which is a function of both pressure and temperature.
High values for the Earth's mantle indicates that convection within the Earth is vigorous and timevarying, and that convection is responsible for almost all the heat transported from the deep interior to the surface.
See also
Notes

^ M. FavreMarinet and S. Tardu, Convective Heat Transfer, ISTE, Ltd, London, 2009

^ ^{a} ^{b} Bunge, HansPeter; Richards, Mark A.; Baumgardner, John R. (1997). "A sensitivity study of threedimensional spherical mantle convection at 10^{8} Rayleigh number: Effects of depthdependent viscosity, heating mode, and endothermic phase change".
References
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