Turbulent Prandtl Number
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The turbulent Prandtl number (Prt) 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 Prt 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, Prt 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 \varepsilon_M and heat transfer \varepsilon_H as
-\overline = \varepsilon_M \frac and -\overline = \varepsilon_H \frac
where -\overline is the apparent turbulent shear stress and -\overline is the apparent turbulent heat flux.
The turbulent Prandtl number is then defined as
\mathrm_\mathrm = \frac. 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:
\bar \frac + \bar \frac = -\frac \frac + \frac \left \nu \frac - \overline) \right
Turbulent thermal boundary layer equation,
\bar \frac + \bar \frac = \frac \left (\alpha \frac - \overline \right). Substituting the eddy diffusivities into the momentum and thermal equations yields
\bar \frac + \bar \frac = -\frac \frac + \frac \left \nu + \varepsilon_M) \frac\right/math>
and
\bar \frac + \bar \frac = \frac \left \alpha + \varepsilon_H) \frac\right
Substitute into the thermal equation using the definition of the turbulent Prandtl number to get
\bar \frac + \bar \frac = \frac \left \alpha + \frac) \frac\right


Consequences

In the special case where 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 ...
and turbulent Prandtl number both equal unity (as in 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 ...
), 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.


References


Books

* {{NonDimFluMech Convection Dimensionless numbers of fluid mechanics Fluid dynamics Heat transfer