Turbulent Prandtl number

Summary

The turbulent Prandtl number (Prt) is a non-dimensional term defined as the ratio between the momentum eddy diffusivity and the heat transfer eddy diffusivity. It is useful for solving the heat transfer problem of turbulent boundary layer flows. The simplest model for Prt is the Reynolds analogy, which yields a turbulent Prandtl number of 1. From experimental data, Prt has an average value of 0.85, but ranges from 0.7 to 0.9 depending on the Prandtl number of the fluid in question.

DefinitionEdit

The introduction of eddy diffusivity and subsequently the turbulent Prandtl number works as a way to define a simple relationship between the extra shear stress and heat flux that is present in turbulent flow. If the momentum and thermal eddy diffusivities are zero (no apparent turbulent shear stress and heat flux), then the turbulent flow equations reduce to the laminar equations. We can define the eddy diffusivities for momentum transfer   and heat transfer   as
  and  
where   is the apparent turbulent shear stress and   is the apparent turbulent heat flux.
The turbulent Prandtl number is then defined as
 

The turbulent Prandtl number has been shown to not generally equal unity (e.g. Malhotra and Kang, 1984; Kays, 1994; McEligot and Taylor, 1996; and Churchill, 2002). It is a strong function of the molecular Prandtl number amongst other parameters and the Reynolds Analogy is not applicable when the molecular Prandtl number differs significantly from unity as determined by Malhotra and Kang;[1] and elaborated by McEligot and Taylor[2] and Churchill [3]

ApplicationEdit

Turbulent momentum boundary layer equation:
 
Turbulent thermal boundary layer equation,
  Substituting the eddy diffusivities into the momentum and thermal equations yields
 
and
 
Substitute into the thermal equation using the definition of the turbulent Prandtl number to get
 

ConsequencesEdit

In the special case where the Prandtl number and turbulent Prandtl number both equal unity (as in the Reynolds analogy), the velocity profile and temperature profiles are identical. This greatly simplifies the solution of the heat transfer problem. If the Prandtl number and turbulent Prandtl number are different from unity, then a solution is possible by knowing the turbulent Prandtl number so that one can still solve the momentum and thermal equations.

In a general case of three-dimensional turbulence, the concept of eddy viscosity and eddy diffusivity are not valid. Consequently, the turbulent Prandtl number has no meaning. [4]

ReferencesEdit

  1. ^ Malhotra, Ashok, & KANG, S. S. 1984. Turbulent Prandtl number in circular pipes. Int. J. Heat and Mass Transfer, 27, 2158-2161
  2. ^ McEligot, D. M. & Taylor, M. F. 1996, The turbulent Prandtl number in the near-wall region for Low-Prandtl-number gas mixtures. Int. J. Heat Mass Transfer., 39, pp 1287--1295
  3. ^ Churchill, S. W. 2002; A Reinterpretation of the Turbulent Prandtl Number. Ind. Eng. Chem. Res. , 41, 6393-6401. CLAPP, R. M. 1961.
  4. ^ Kays, W. M. (1994). "Turbulent Prandtl Number—Where Are We?". Journal of Heat Transfer. 116 (2): 284–295. doi:10.1115/1.2911398.

BooksEdit

  • Kays, William; Crawford, M.; Weigand, B. (2005). Convective Heat and Mass Transfer, Fourth Edition. McGraw-Hill. ISBN 978-0-07-246876-2.