Page 292 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

262 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

model for the thermal field. As for subgrid models for the velocity field, there are
many variants, which have been envisaged and proposed. The reader may refer to
Sagaut (2006) and Pope (2000) to have an overview. Among them, there is obviously
the eddy heat diffusivity approach (Eq. 6.1.2.53), which is the analog of the eddy-
viscosity concept. In this case, one may build a model for a subgrid Prandtl number
(Eq. 6.1.2.52) using, in the simplest case, a constant value (as mentioned previously),
or using a dynamic procedure based on the Germano identity. Such models have been
proposed by Moin et al. (1991) and Wong and Lilly (1991), and very lastly revisited by
Li (2016). These models show stability issues and need specific treatment to be used
(clipping, averaging, etc., see Sagaut, 2006). An other way is to directly express a
model for the eddy heat diffusivity α SGS . For the case of liquid metals, Gr€ otzbach
(1981) proposed a self-adaptive heat flux model where the eddy heat diffusivity is
modeled by a Prandtl energy length scale assumption.
However, in Section 6.1.2.4, the simulations performed at low Prandtl numbers are
well-resolvedLES,forexample,thegridcutoffwas chosensuchthatthethermal fieldis
fully resolved by the LES grid, as proposed in Fig. 6.1.2.4. Those simulations can be
called V-LES/T-DNS and do not use any subgrid model for the subgrid-scale heat flux.

6.1.2.4 Application

6.1.2.4.1 The turbulent channel flow
The application given in this chapter is the developed turbulent flow of a low Prandtl
number fluid through a uniformly heated channel. It is numerically tackled by the
computation of a time-developing flow between two plates with periodic boundary
conditions in the streamwise and spanwise directions (Fig. 6.1.2.5) and which reaches
statistical equilibrium.
The flow is characterized by the Reynolds number based on the friction velocity u τ :
u τ δ 2
Re τ ¼ ν where u τ is related to the wall shear stress: u ¼ τ w =ρ, δ is half the channel
τ
width and ν is the kinematic viscosity. In order to handle periodic boundary condi-
tions, it is required to use a modified temperature θ(x, y, z, t) variable such that

dT w
T ¼ x θ, (6.1.2.58)
dx
where the wall temperature gradient forcing compensates the temperature increase in
the periodic streamwise direction due to the constant mean heat flux q g (the acco-
f
w
lade sign stands for the mean value along the wall surface):
dT w f q g
w
¼ , (6.1.2.59)
dx ρ c δ u hi
where u hi is the streamwise time and space averaged velocity (the brace sign stands
for the mean value along the channel cross-section). This leads to a modified energy
equation for θ with the following source term: S θ ¼ u dT w :
dx