Page 280 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

250 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

6.1.2.2.2 Implicitly filtered LES or grid-LES

Applying to the set of governing equations the grid projection operator ð…Þ, and
g
assuming that the filter operator commutes with the derivative operator, we obtain
the filtered equations:

∂~ u i
¼ 0, (6.1.2.10)
∂x i

∂
∂~ u i ∂~ u i ~ u j
g
+ ¼ Pδ ij +2νS ij : (6.1.2.11)
e
∂t ∂x j ∂x j
The commutation is verified as far as the filter is homogeneous (Sagaut, 2006); how-
ever, when dealing with nonhomogeneous grid, this property is not necessary fulfilled
and commutation errors should be accounted for, as proposed in Ghosal and Moin
(2001) and explained in Sagaut (2006). The nonlinear term of the momentum
(Eq. 6.1.2.7) gives a filtered product of DNS variables (Eq. 6.1.2.11). Indeed, in this
LES frame, only the filtered variables are available (i.e., those projected on the LES
grid). This term is at the source of subgrid-scale stress. Indeed, in order to recover the
filtered product of computable variables (e.g., filtered variables), we need to introduce
a new term on the RHS:

∂~ u i ∂~ u i ~ u j ∂ SGS
g
+ ¼ Pδ ij +2νS ij + σ ij , (6.1.2.12)
∂t ∂x j ∂x j
where the subgrid-scale stress tensor is defined as:

σ SGS ¼H u i , u j ¼ ~ u i ~ u j g u j , (6.1.2.13)
u i
g
ij

with the operator Ha,bÞ ¼ ~ ab ab, and it must to be modeled and represents the
ð
effect of the scales not captured by the grid (lost information). The modeling of this
term is crucial in large-eddy simulation, because it allows the final dissipation of the
energy cascade. However, if this term is omitted, the simulation blows up as nothing
dissipates the energy cascading from the large scales.
6.1.2.2.3 Closure problem

6.1.2.2.3.1 Practical LES equation set

In Eq. (6.1.2.7), S ij is a traceless tensor. We can rewrite the subgrid-scale stress tensor
as the sum of a deviatoric and nondeviatoric part as:

SGS SGS 1 SGS
τ ¼ σ σ δ ij , (6.1.2.14)
ij ij kk
3