In fluid mechanics, the Rayleigh number (Ra, after Lord Rayleigh[1]) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection.
Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection.
The Rayleigh number is defined as the product of the Grashof number (Gr), which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number (Pr), which describes the relationship between momentum diffusivity and thermal diffusivity: Ra = Gr × Pr.
Gravity causes denser parts of the fluid to sink, which is called convection.
Lord Rayleigh studied[2] the case of Rayleigh-Bénard convection.
[6] When the Rayleigh number, Ra, is below a critical value for a fluid, there is no flow and heat transfer is purely by conduction; when it exceeds that value, heat is transferred by natural convection.
[3] 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
time scale for thermal transport via convection at speed
{\displaystyle \mathrm {Ra} ={\frac {\text{time scale for thermal transport via diffusion}}{{\text{time scale for thermal transport via convection at speed}}~u}}.}
in all three dimensions[clarification needed] and mass density difference
From the Stokes equation, when the volume of fluid is sinking, viscous drag is of the order
The time scale for thermal diffusion across a distance
For free convection near a vertical wall, the Rayleigh number is defined as:
where: In the above, the fluid properties Pr, ν, α and β are evaluated at the film temperature, which is defined as:
For a uniform wall heating flux, the modified Rayleigh number is defined as:
where: 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.
where: 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.
[8] 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.
[9] 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.
In a bulk fluid, i.e., not in a porous medium, from the Stokes equation, the falling speed of a domain of size
In porous medium, this expression is replaced by that from Darcy's law
This also applies to A-segregates, in the mushy zone of a solidifying alloy.
[8] 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:
where: A Rayleigh number for bottom heating of the mantle from the core, RaT, can also be defined as:
where: 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.