Page 282 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

252 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

SGS SGS

2
with P¼ P=ρ + K SGS and τ the deviatoric part of σ ¼ ~ u i ~ u j u i u j to be modeled.
g
3 ij ij
6.1.2.2.3.2 The closure problem
0
By using the scale separation operator ϕ ¼ϕ + ϕ , we can rewrite the effective
subgrid-scale stress as:

h i
SGS
σ ¼ ~ u i ~ u j g u j ¼ g 0 g 0 + u u ¼ C ij + R ij : (6.1.2.20)
~ u i u + ~ u j u
g 0
0
g
ij u i j i i j
The first term is called the cross-term and represents a truncated product between
resolved scales and unresolved scales. The second term is the Reynolds term by anal-
ogy of its counterpart in RANS closure problem. We will see in Section 6.1.2.3 that
both terms are usually modeled, at least their deviatoric part, as a whole even though
more complex models attempt to model them separately (see Thiry and
Winckelmans, 2016).
It is worth to mention here a fundamental difference between the RANS and LES
approaches. To obtain the RANS equations, we define an averaging operator which is
actually a time average and defined as:

1 Z t + Δt=2
hϕi¼ ϕdt, (6.1.2.21)
Δt t Δt=2

0
ϕ ¼hϕi + ϕ , (6.1.2.22)
0
where hϕi is the field actually resolved by the RANS equations and ϕ is the fluctu-
ating part which is not resolved. When this operator is applied to the governing equa-
tions, the RANS equations are obtained and are very similar to LES equations but they
are written for (time) averaged quantities hϕi. The unknown terms are also understood
as a stress tensor, called the Reynolds stress tensor, and can be written as:

h i
hσ RANS hhu i iu i + hhu j iu i + hu u i ¼hC ij i + hR ij i¼hR ij i, (6.1.2.23)
0 0
0
0
ij i¼ j i i j
where the major difference with LES is that the cross-term does not exist as a property
of the averaging operator is hhϕiϕ i¼ 0, contrary to the same operation done with
0
a filter.
6.1.2.2.4 Heat transfer in LES

If we consider the fluid incompressible with constant thermophysical properties, the
energy equation can be cast in a conservation equation for the temperature, which acts
as a passive scalar. However, when dealing with natural convection, the temperature
becomes an active scalar because a direct coupling is observed between the