Turbulent Prandtl number

The turbulent Prandtl number (Pr_t) 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 Pr_t is the Reynolds analogy, 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 of the fluid in question.

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 ${\displaystyle \varepsilon _{M}}$ and heat transfer ${\displaystyle \varepsilon _{H}}$ as
${\displaystyle -{\overline {u'v'}}=\varepsilon _{M}{\frac {\partial {\bar {u}}}{\partial y}}}$ and ${\displaystyle -{\overline {v'T'}}=\varepsilon _{H}{\frac {\partial {\bar {T}}}{\partial y}}}$
where ${\displaystyle -{\overline {u'v'}}}$ is the apparent turbulent shear stress and ${\displaystyle -{\overline {v'T'}}}$ is the apparent turbulent heat flux.
The turbulent Prandtl number is then defined as
${\displaystyle \mathrm {Pr} _{\mathrm {t} }={\frac {\varepsilon _{M}}{\varepsilon _{H}}}.}$

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