Page 288 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

258 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

^ ^ : (6.1.2.43)
d
g
^ ^ 2 ^ 2
2 ^ 2 d
^ α ij ¼ 2α Δ jSjS ij ¼ 2α Δ jSjS ij
Numerical tests have showed that α ¼ 2 (e.g., using a test filter having twice the size of
the original filter) is a good value (Sagaut, 2006; Pope, 2000). The application of
Eq. (6.1.2.41) allows then to compute the Smagorinsky constant of Eq. (6.1.2.27).

6.1.2.3.3 Multiscale models
6.1.2.3.3.1 The approach

Multiscale models have been proposed as an improvement of classical eddy-viscosity
models because those tend to dissipate too much the largest scale, which are well
resolved by the mesh (as it is the purpose of an LES). Hughes et al. (2001) proposed
a model, which only acts on the high wavenumber range of the resolved field (i.e., on
the smallest resolved scales). Indeed, as the subgrid-scale stress is the effect of the
unresolved part of the spectrum over the resolved part, one might think that this inter-
action essentially affects the resolved eddies, which are close to the wavenumber cut-
off ( Jeanmart, 2002). For separating the smallest resolved scale from the large ones, it
is then necessary to filter the LES field using a high-pass filter:

s
ϕ ¼ϕ ϕ , (6.1.2.44)

s
where ϕ is the small part of the LES field; ϕ and ϕ is the low-pass filtered LES field.
In Hughes et al. (2001) the distinction between the small-scale velocity field and the
large-scale velocity field is sharp, because a sharp Fourier cutoff was used to filter the
LES field with a cutoff at k c /2. The small field was then considered between k c /2 and
k c . Those models are called variational multiscale models. Later, Jeanmart and
Winckelmans (2007) proposed a regularized version by using a smooth discrete filter.
By using this approach, we can use a smooth discrete filter iterated n times to reach the
necessary order (2n):

n
n
n h i o
2 2 n 2
¼ I δ =4 I δ =4 I δ =4 u , (6.1.2.45)
u n x y z
2
where I is the identity operator and fδ gf i, j,k ¼ f i +1, j,k 2f i, j,k + f i 1, j,k on a uniform
x
grid (see Jeanmart and Winckelmans, 2007 for the case of nonuniform grids). The cho-
sen low-pass filter is here a discrete Gaussian filter iterated n times (order 2n). Once
the small field is obtained, a (small) strain rate tensor based on the resolved small field
can be expressed:

s 1 ∂~ u s i ∂~ u s
j
S ¼ + : (6.1.2.46)
ij
2 ∂x j ∂x i
It is now possible to define different classes of eddy-viscosity models according to the
use of the small-velocity field to express the strain tensor and/or the subgrid viscosity.