Page 286 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

256 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Fig. 6.1.2.3 Resolved (by LES) and modeled (SGS) parts of a spectrum for two different levels
of filtering.

SGS,1
¼ σ ^ ij + ^ SGS,2 ~ u i , ~ u j , (6.1.2.34)
σ
ij
SGS,1
^
¼ σ ^ ij + L ij , (6.1.2.35)
which means that the residual stress of the double-filtered equations is the level 1 sub-
grid stress tensor filtered at the second level, plus the level 2 subgrid stress tensor
expressed by using the level 1 velocity field. In that sense the latter term is not a sub-
^ ^
grid stress tensor but a Leonard tensor term (L ij ¼ ~ u i ~ u j ~ u i ~ u j ) as it is determined from
the resolved field (known quantity). Another interesting property, as far as a sharp
Fourier cutoff is considered, is the fact that filtering the level 1 LES by a level 2 filter

d
(larger than filter 1) is similar to performing a level 2 LES (i.e., ð…Þ¼ ð…Þ), thus:
g
d
SGS,1
^
σ
^ ¼ ^ SGS,2 ¼ σ ^ + L ij : (6.1.2.36)
σ ij ij ij
Eq. (6.1.2.35)or (6.1.2.36) is called the Germano identity (Germano et al., 1991) and
is general. It gives a relation between two subgrid stress tensors at the two filter levels.
The Leonard tensor represents a stress tensor between the two cutoffs and can be com-
puted without any model from the resolved field.
Because the Leonard term can determined, the idea is to ensure that the Germano
identity is verified for the SGS models. In this regard, the error or the residue is cal-
culated as:

SGS,1
^
E ij ¼ σ ^ ij ^ SGS,2 + L ij : (6.1.2.37)
σ
ij