Rayleigh number

Classical Definition

For free convection near a vertical wall, the Rayleigh number is defined as:

R a x = g β ν α ( T s − T ∞ ) x 3 = G r x P r

where:

x is the characteristic lengthRa_x is the Rayleigh number for characteristic length xg is acceleration due to gravityβ is the thermal expansion coefficient (equals to 1/T, for ideal gases, where T is absolute temperature). ν is the kinematic viscosityα is the thermal diffusivityT_s is the surface temperatureT_∞ is the quiescent temperature (fluid temperature far from the surface of the object)Gr_x is the Grashof number for characteristic length xPr 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:

T f = T s + T ∞ 2 .

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

R a x ∗ = g β q o ″ ν α k x 4

where:

q"_o is the uniform surface heat fluxk is the thermal conductivity.

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

Other Definitions

The Rayleigh number can be also 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:

R a = Δ ρ ρ 0 g K ¯ L α ν = Δ ρ ρ 0 g K ¯ R ν

where:

K is the mean permeability (of the initial portion of the mush)L is the characteristic length scaleα is the thermal diffusivityν is the kinematic viscosityR is the solidification or isotherm speed.

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. 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.

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:

R a H = g ρ 0 2 β H D 5 η α k

where:

H is the rate of radiogenic heat production per unit massη is the dynamic viscosityk is the thermal conductivityD is the depth of the mantle.

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

R a T = ρ 0 2 g β Δ T s a D 3 C P η k

where:

ΔT_sa is the superadiabatic temperature difference between the reference mantle temperature and the core–mantle boundaryC_P is the specific heat capacity at constant pressure.

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.