Page 283 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Large-eddy simulation: Application to liquid metal fluid flow and heat transfer 253

momentum equation and the energy equation via the buoyancy term. For the sake of
simplicity, we will only deal here with a passive scalar equation. In this frame, the
energy equation can be written in the temperature form as:

!
∂T e ∂T ~ u j ∂ ∂ T SGS
f
e
+ ¼ α + τ T , (6.1.2.24)
∂t ∂x j ∂x j ∂x j
SGS

where the subgrid heat flux, defined as τ ¼ T ~ u j Tu j , needs to be modeled, and
g
f
T
where α is the molecular heat diffusivity. The dynamics of the temperature can follow
different regimes depending on how it is coupled with the flow dynamics. The param-
eter driving this coupling is the Prandtl number of the fluid (or the Schmidt number if
the scalar represents the transport of a specie):
ν
Pr ¼ : (6.1.2.25)
α
Liquid metals are characterized by a very low molecular Prandtl. This specificity
brings an opportunity of simplification to deal with the subgrid-scale term as will
be explained in Section 6.1.2.3.4.2 and practically used in Bricteux et al. (2012)
and Duponcheel et al. (2014).

6.1.2.3 Subgrid-scale models

6.1.2.3.1 Eddy-viscosity models
Most of the subgrid-scale models used in engineering applications rely on the eddy-
viscosity concept, by analogy to the approach adopted in RANS. This concept relates
the subgrid stress tensor in Eq. (6.1.2.19) to the resolved strain rate tensor, thus assum-
ing both tensors are aligned via a parameter, the subgrid-scale viscosity ν SGS , by:

SGS
τ ij ¼ 2ν SGS S ij : (6.1.2.26)
Contrary to the RANS approach for the turbulent viscosity, in LES, the subgrid-scale
viscosity model is supposed to have less impact on the accuracy of the results: because
most of the energy cascade is resolved, the subgrid-scale model should only act for the
final removal of the energy cascading from the resolved scales. Indeed, the dissipation
rate E is fixed by the larger scales, which are supposed to be mostly solved by an LES
contrary to an RANS. Consequently, it is expected that even a simple subgrid-scale
model, still playing his dissipation role, could do the job without altering the correct-
ness of the simulation; this is a major advantage of LES over RANS. Thus, most of the
subgrid-scale models used in LES are based on algebraic models rather than on solv-
ing additional transport equations as it is the case in RANS, even though some similar
approaches have also been developed (see Sagaut, 2006; Pope, 2000) for LES.