Page 281 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Large-eddy simulation: Application to liquid metal fluid flow and heat transfer 251

SGS
where σ is related to the subgrid-scale turbulent kinetic energy:
kk

σ SGS ¼ g u k ~ u k ~ u k ¼ 2K SGS : (6.1.2.15)
u k
g
kk
In this frame the final form of the momentum equation is:
∂~ u i ∂~ u i ~ u j ∂ SGS
g
+ ¼ Pδ ij +2νS ij + τ ij , (6.1.2.16)
∂t ∂x j ∂x j
2 SGS
where the effective pressure is defined as P¼P + K . Thus the subgrid modeling
3
purpose is to propose a model for τ SGS , that is, for the deviatoric part of
ij
SGS
σ ¼ ~ u i ~ u j g u j . In some advanced subgrid-scale models, the subgrid turbulent
ij u i
g
kinetic energy is modeled separately.
This formulation (Eq. 6.1.2.16) is different to what it is possible to find in literature,
for example:
∂
∂~ u i ∂~ u i ~ u j SGS
+ ¼ Pδ ij +2νS ij + τ ij , (6.1.2.17)
∂t ∂x j ∂x j
SGS
SGS
with τ is the deviatoric part of σ ¼ ~ u i ~ u j g u j . This formulation is inadequate for
ij ij u i
a grid-LES, because all the terms can be solely capture on the LES grid. Indeed, the
product ~ u i ~ u j contains higher frequencies than ~ u i or ~ u j alone, and would require a finer
mesh to be represented. The form of Eq. (6.1.2.17) could be nevertheless obtained if
the scale separation process used an explicit filtering only (see Sagaut, 2006), this
approach could be called mathematical LES. In this case, the residual stress appearing
in the RHS of Eq. (6.1.2.17) would be called a subfilter stress. This subfilter stress
could be then determined by partial deconvolution (see Yeo, 1987; Leonard, 1974;
Carati et al., 2001). Thus the effective subgrid-scale stress in a grid-LES is not a com-
mutator operator as it would be the case for the subfilter stress, and requires to be
modeled. Moreover, if an explicit LES formulation is solved on a computational grid,
it is equivalent to a double filtering operation, one explicit filter and one Fourier cutoff
for the projection of the explicitly filtered equations on the grid. The result for the RHS
of the momentum equation will be a residual stress composed of a subfilter stress
(deconvolvable) and a subgrid-scale stress (requiring a model).
Consequently, the final formulation to keep is:

∂~ u i
¼ 0, (6.1.2.18)
∂x i

∂~ u i ∂~ u i ~ u j ∂ SGS
g
+ ¼ Pδ ij +2νS ij + τ ij , (6.1.2.19)
∂t ∂x j ∂x j