The turbulent Prandtl number (Pr
t) is a
non-dimensional
Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
term defined as the ratio between the momentum
eddy diffusivity and the heat transfer eddy diffusivity. It is useful for solving the
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
problem of turbulent boundary layer flows. The simplest model for Pr
t is the
Reynolds analogy The Reynolds Analogy is popularly known to relate turbulent momentum and heat transfer.Geankoplis, C.J. ''Transport processes and separation process principles'' (2003), Fourth Edition, p. 475. That is because in a turbulent flow (in a pipe or in a ...
, which yields a turbulent Prandtl number of 1. From experimental data, Pr
t has an average value of 0.85, but ranges from 0.7 to 0.9 depending on the
Prandtl number
The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number is given as:where:
* \nu : momentum d ...
of the fluid in question.
Definition
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
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
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; and elaborated by McEligot and Taylor and Churchill
Application
Turbulent momentum boundary layer equation:
Turbulent thermal boundary layer equation,
Substituting the eddy diffusivities into the momentum and thermal equations yields