Page 284 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

254 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

The subgrid-scale viscosity has obviously the same dimensions as the molecular
2 1
kinematic viscosity (e.g., m s ). By dimensional analysis, it required a time scale
T and a length scale L; this is the basis of any subgrid-scale viscosity model.

6.1.2.3.1.1 The Smagorinsky model

The Smagorinsky model (Smagorinsky, 1963) is considered as the pioneer subgrid-
scale model for LES. For this model, the length scale is chosen as a grid-
1=3
characteristic length scale or filter-width length scale, L¼ Δ .The
¼ h x h y h z
q ffiffiffiffiffiffiffiffiffiffiffiffiffi
1
time scale is defined from the norm of the strain rate tensor T ¼ 2S ij S ij .This
finally gives:
1=2

ν SGS ¼ C S Δ 2S ij S ij ¼ C S Δ j S j, (6.1.2.27)
2
2
where C S is the Smagorinsky constant. As C s is real and positive, this model is purely
dissipative and does not account for backscatter (inverse energy cascade). Lily
(1967) established the following relation to calculate the constant for high Reynolds
flows:

3=2
1 2
C S ¼ , (6.1.2.28)
π 2 3C K
where C K is the Kolmogorov constant; for C K ¼ 1.6, C S ¼ 0.027. For moderate Reyn-
olds flows, this value of C S is too high, which provides over-dissipation involving
relaminarization of the flow in such regions. Furthermore, this model suffers from
an incorrect behavior as the subgrid-scale viscosity does not tend to zero as
approaching the wall. This issue can be solved by using a damping function proposed
by Piomelli et al. (1993), which also provides the correct asymptotic behavior, for
+3
example, in Oðy Þ (Bricteux, 2008):
3
ν SGS ¼ C S Δ 1 e ðy =25Þ j Sj: (6.1.2.29)
+
2

6.1.2.3.1.2 The WALE model

The WALE (Wall-Adapting Local Eddy-viscosity) model is an eddy-viscosity model
developed by Nicoud and Ducros (1999), which is designed to provide a correct
+3
Oðy Þ near-wall scaling for the eddy viscosity and to be inactive in a pure laminar
shear flow. In this frame, the subgrid stress model is given by:
SGS
τ ¼ 2ν SGS S ij , (6.1.2.30)
ij