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  is the apparent turbulent shear stress and  is the apparent turbulent heat flux.
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