Respectively, where [mu]t is the turbulent molecular viscosity, [sigma]t is the constant of turbulent prandtl number
and [rho][gamma] is the prandtl number nanofluids.
This can be often attributed to an improper near-wall turbulence modeling and the deficiency of the wall heat transfer models (based on the so called P-functions) that do not properly account for the variation of the turbulent Prandtl number
in the wall proximity, [see e.g.
The diffusion coefficient of heat (KH) is defined as [K.sub.H] = [K.sub.M]/Pr, where Pr is the turbulent Prandtl number
, assumed to be one for simplicity, and [K.sub.M] = [l.sub.m] [square root of ([[alpha].sub.0]E)] is the diffusion coefficient of momentum.
while the eddy diffusivity appearing in the energy equation is [[gamma].sub.T] = [v.sub.T]/[[sigma].sub.T], [[sigma].sub.T], being the turbulent Prandtl number
. The production and stratification terms in the turbulence model are given by:
The simple first order central difference version of the energy equation is expressed as the local product of the Reynolds number and the turbulent Prandtl number
(RePr = Peclet number) using the effective viscosity of the central node:
In this paper, turbulent flow computations have been made using the low-Reynolds number k - [epsilon] turbulence model proposed by Launder-Sharma  with turbulent heat transfer represented using a turbulent Prandtl number
, [Pr.sub.t], equal to 0.9.
where [sigma.sub.t]=0.6 stands for the turbulent Prandtl number
In addition, they concluded that the predictive ability of the k-[epsilon] models is expected to improve when turbulent Prandtl number
variations near the wall are included.
where [Pr.sub.T] = 0.7 is the turbulent Prandtl number
, and [[mu].sub.T] is the turbulent viscosity, given by:
Where [Pr.sub.t], ranging from 0.8 to 1.3 in this study, is the turbulent Prandtl number
. The five empirical constants of Eqs.
The water film Reynolds number related to the film thickness markedly exerted an influence on the eddy viscosity and the turbulent Prandtl number
of the water film.
[[sigma].sub.k], [[sigma].sub.[epsilon]] and aT are the turbulent Prandtl number
for k, [epsilon], and T, respectively.