Advanced SGS models The Smagorinsky model is purely dissipative: the direction of energy flow is

3.8.4 Advanced SGS models The Smagorinsky model is purely dissipative: the direction of energy flow is

exclusively from eddies at the resolved scales towards the sub-grid scales. Leslie and Quarini (1979) have shown that the gross energy flow in this direction is actually larger and offset by 30% backscatter – energy transfer in reverse direction from SGS eddies to larger, resolved scales. Furthermore, analysis of results from direct numerical simulation (DNS) by Clark et al. (1979) and McMillan and Ferziger (1979) revealed that the correlation between the actual SGS stresses (as computed by accurate DNS) and the modelled SGS stresses using the Smagorinsky–Lilly model is not particularly strong. These authors came to the conclusion that the SGS stresses should not be taken as proportional to the strain rate of the whole resolved flow field, but, in recogni- tion of the actual energy cascade processes (see section 3.1), should be estimated from the strain rate of the smallest resolved eddies. Bardina et al. (1980) pro- posed a method to compute local values of C SGS based on the application of

two filtering operations, taking the SGS stresses to be proportional to the stresses due to eddies at the smallest resolved scale. They proposed

τ ij = ρC′( R i R j − X i X j )

where C ′ is an adjustable constant and the factor in the brackets can be eval- uated from twice-filtered resolved flow field information. The correlation between the actual SGS stresses as computed with a DNS and the modelled SGS stresses was found to be much improved, but the appearance of nega- tive viscosities generated stability problems. They proposed adding a damp- ing term with the form of the Smagorinsky model (3.94)–(3.95) to stabilise the calculations, which yields a mixed model:

(3.99) The value of the constant C ′ depends on the cutoff width used for the sec-

τ ij = ρC′( R i R j − X i X j ) −2 ρC 2 SGS ∆ 2 |D|D ij

ond filtering operation, but is always close to unity. Germano (1986) proposed a different decomposition of the turbulent stresses. This formed the basis of the dynamic SGS model (Germano et al., 1991) for the computation of local values of C SGS . In Germano et al.’s

decomposition of turbulent stresses the difference of the SGS stresses for two different filtering operations with cutoff widths ∆ 1 and ∆ 2 , respectively, can be evaluated from resolved flow data:

(3.100) The bracketed superscripts (1) and (2) indicate filtering at cutoff widths ∆ 1

τ (2) ij − τ (1) ij = ρL ij ≡ ρ R i R j − ρ X i X j

and ∆ 2 . The SGS stresses are modelled using Smagorinsky’s model (3.94) – (3.95) assuming that the constant C SGS is the same for both filtering operations. It can be shown that this yields:

L ij −L kk δ ij =C 2 SGS M ij

(3.101a)

(3.101b) Lilly (1992) suggested a least-squares approach to evaluate local values of

with M ij = −2∆ 2 2 | Y |Y ij + 2∆ 2 1 | D |D ij

C SGS :

CHAPTER 3 TURBULENCE AND ITS MODELLING

The angular brackets 〈 〉 indicate an averaging procedure. As Bardina et al. (1980) before them, Germano et al. (1991) found that the dynamic SGS model yielded highly variable eddy viscosity fields including regions with negative values. This problem was resolved by averaging: for problems with homogeneous directions (e.g. two-dimensional planar flows) the averaging takes place over the homogeneous direction; in complex flows an average over a small time interval is used.

Germano (in Peyret and Krause, 2000) reviewed other formulations for the dynamic calculation of the SGS eddy viscosity. The dynamic model and other advanced SGS models are reviewed in Lesieur and Métais (1996) and Meneveau and Katz (2000). The interested reader is referred to these publi- cations for further material in this area.