Page 285 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Large-eddy simulation: Application to liquid metal fluid flow and heat transfer 255

3=2
d d
S S
ij ij
ν SGS ¼ C W Δ , (6.1.2.31)
2
5=4
5=2 d d

S ij S ij + S S
ij ij
where

1 ∂~ u i ∂~ u k ∂~ u j ∂~ u k
¼ + ¼ S ik S kj + Ω ik Ω kj (6.1.2.32)
S ij
2 ∂x k ∂x j ∂x k ∂x i
gives the correct near-wall behavior. The tensor Ω is the antisymmetric part of the
velocity gradient tensor. The model coefficient was calibrated by Nicoud and
Ducros (1999) on homogeneous isotropic turbulence at moderate Reynolds and it
gives C W ¼ 0.25. Bricteux et al. (2009) calibrated the coefficient to C W ¼ 0.20 for
their channel flow (at Re τ ¼ 395) and showed that the model is inappropriate for lam-
inar vortical flows because it is still active in solid body rotation regions. They con-
cluded that the WALE model could be too dissipative, although less than
Smagorinsky, when solid rotation takes place. This was the rationale for proposing
a new multiscale version of the WALE model, which will be presented in
Section 6.1.2.3.3.2.

6.1.2.3.2 Dynamic models

6.1.2.3.2.1 The approach
The philosophy of the dynamic approach, in general, is to calculate, locally, the coef-
ficient of the subgrid stress model during a simulation. For example, this procedure
allows to locally determine C S (Eq. 6.1.2.27) in the Smagorinsky model, but could
be applied to any model so almost each subgrid-scale model has its dynamic version,
yet it does not work well for the WALE model. This approach was first introduced by
Germano et al. (1991) and relies on the application of a second-level filtering having a
larger filter width than the LES filter. This second-level filter is also called a test filter.
As we here deal with practical implicitly filtered LES, the second filter can be con-
structed as the projection on a larger grid size αh with α > 1. In spectral space, it means
we apply a second sharp Fourier cutoff k 2 < k 1 , k 1 being the first-level cutoff of the
LES (see Fig. 6.1.2.3).
Eq. (6.1.2.7), using a projection operator ð…Þ, is considered as the level 1 LES. The
g
level 2 (only) LES equation could be obtained by replacing the operator ð…Þ by the
g
operator ð…Þ related to the level 2 wave number cutoff k 2 ¼ π/αh. Hence it is possible
d
to obtain a relation between the different stress tensors:
SGS,1
d
^ ¼ σ ^ ^ ^ g, (6.1.2.33)
d
g
σ ij ij + ~ u i ~ u j ~ u i ~ u j