Rayleigh number

In fluid mechanics, the Rayleigh number (, after Lord Rayleigh) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection. For most engineering purposes, the Rayleigh number is large, somewhere around 10⁶ to 10⁸.

The Rayleigh number 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: . 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.

<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

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.

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.

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. 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 <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.

A Rayleigh number for bottom heating of the mantle from the core, Ra_T, 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:

• ΔT_sa 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_P is the specific heat capacity at constant pressure.