Page 317 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Turbulent heat transport 287

υ t
as Pe t ¼ Pr ð = Þ. It is worth mentioning that this property is locally available from
υ
the RANS models. Nevertheless, this correlation is given as
0:7
Pr t ¼ 0:85 + (6.2.1.13)
Pe t
Following the work of Kays (1994), Weigand et al. (1997) proposed a correlation:

1 1 1 2 1
ð
p
¼ + CPe t ffiffiffiffiffiffiffiffiffiffi CPe t Þ 1 exp p ffiffiffiffiffiffiffiffiffiffi (6.2.1.14)
Pr t 2Pr t∞ Pr t∞ CPe t Pr t∞
where C¼ 0.3 and the approximate value for Pr t far from the wall is given as
100
Pr t∞ ¼ 0:85 + (6.2.1.15)
PrRe 0:888
It is explicit from the formulation that this correlation mixes both local and the global
parameters. In the end, it gives a local value as a function of the global quantities. This
is mainly because of the Re number dependency. Duponcheel et al. (2014) did a thor-
ough assessment of these aforementioned correlations and compared the results with
their high fidelity simulations of channel flows at Re τ ¼590 for two different Pr num-
bers, that is, 0.01 and 0.025 (see Fig. 6.2.1.9).
From Fig. 6.2.1.9, it is clear that among these correlations, the Reynolds correlation
overpredicts the Pr t . On the other hand, the Kays correlation yields the best results for
y+>100, that is, in the log layer, whereas the correlation of Weigand gives a better
approximation for the near-wall profile of the Pr t . However, the correlation of
Weigand does not clearly follow the reference data profile. Nonetheless, the Kays cor-
relation turns out to be the most attractive one because it provides the best compromise
between accuracy and the simplicity.
Following the Kays correlation, Duponcheel et al. (2014) studied the near-wall
behavior of temperature and derived a wall function for the temperature profile at
low Pr number, which is given as

+ Pr t k +
θ ¼ log 1 + Pry (6.2.1.16)
k Pr t
Following the hypothesis that the molecular and the turbulent diffusivities can be
neglected in the near-wall region, authors assumed that the turbulent diffusivity is lin-
ear. Eventually, an analytic wall function was obtained with a smooth transition from a
linear profile to a logarithmic profile, which is called the mixed law of the wall
(Duponcheel et al., 2014). This mixed law is valid for the whole near-wall region.
It is worth reminding that this mixed law is similar to the proposal of Wolters
(2002), who in fact neglected the turbulent diffusivity in the near-wall region. The
authors used this mixed wall of the wall in combination with the Kays correlation
and obtained results that are shown in Fig. 6.2.1.10. It is clearly noticeable that the