Rayleigh number

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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.

Classical Definition[edit]

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)
  • Rax = Rayleigh number at position x
  • Grx = Grashof number at position x
  • Pr = Prandtl number
  • g = acceleration due to gravity
  • Ts = 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 106 to 108.


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/m2)
  • k = the thermal conductivity (W/m•K)

Other Definitions[edit]

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 [2]


\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}

where

  •  \bar{K} = mean permeability of the initial portion of the mush)
  •  L = characteristic length scale
  •  \alpha = thermal diffusivity
  •  \nu = kinematic viscosity
  •  R = 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.[3] in Prof. Beckermann’s research group[4] showed that for steel alloys the critical Rayleigh number is 17. Pickering et. al [5] explored Torabi Rad’s criterion, and further verified its effectiveness. The critical Rayleigh number for Lead-Tin and Nickel based super-alloys have been also developed in the same research group.

Geophysical applications[edit]

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

\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.[6]

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

\mathrm{Ra}_T = \frac{\rho_{0}g\beta\Delta T_{sa}D^3 c}{\eta k} [6] Where ΔTsa 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 time-varying, and that convection is responsible for almost all the heat transported from the deep interior to the surface.

See also[edit]

Notes[edit]

  1. ^ M. Favre-Marinet and S. Tardu, Convective Heat Transfer, ISTE, Ltd, London, 2009
  2. ^ 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. 
  3. ^ 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. 
  4. ^ "Solidification Lab at the University of Iowa". Solidification Lab. 
  5. ^ Pickering, EJ; Al-Bermani, S; Talamantes-Silva, J (2014). "Application of criterion for A-segregation in steel ingots". Materials Science and Technology. 
  6. ^ 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[edit]

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