Smagorinksy---Lilly SGS model In simple flows such as two-dimensional thin shear layers the Boussinesq

3.8.2 Smagorinksy---Lilly SGS model In simple flows such as two-dimensional thin shear layers the Boussinesq

eddy viscosity hypothesis (3.33) was often found to give good predictions of Reynolds-averaged turbulent stresses. In recognition of the intimate connec- tion between turbulence production and mean strain, the hypothesis takes the turbulent stresses to be proportional to the mean rate of strain. Success of the approach requires that (i) the changes in the flow direction should be slow so that production and dissipation of turbulence are more or less in bal- ance and (ii) the turbulence structure should be isotropic (or if this is not the case the gradients of the anisotropic normal stresses should not be dynami- cally active). Smagorinsky (1963) suggested that, since the smallest turbulent eddies are almost isotropic, we expect that the Boussinesq hypothesis might provide a good description of the effects of the unresolved eddies on the resolved flow. Thus, in Smagorinsky’s SGS model the local SGS stresses R ij are taken to be proportional to the local rate of strain of the

resolved flow D ij = – 1 2 ( ∂ R i / ∂x j + ∂ R j / ∂x i ):

3.8 LARGE EDDY SIMULATION 103

1 A ∂ R i ∂ R j D 1 R ij = −2 µ SGS D ij +R ii δ ij =− µ SGS

+R ii δ ij (3.93)

3 B C E ∂x j ∂x i F 3

The constant of proportionality is the dynamic SGS viscosity µ

1 SGS has dimensions Pa s. The term , which –

3 R ii δ ij on the right hand side of equation (3.93) performs the same function as the term − – 2 3 ρkδ ij in equation (3.33): it

ensures that the sum of the modelled normal SGS stresses is equal to the kinetic energy of the SGS eddies. In much of the LES research literature the above model is used along with approximate forms of the Leonard stresses L ij and cross-stresses C ij for the particular filtering function applied in the work.

Meinke and Krause (in Peyret and Krause, 2000) review applications of finite volume/LES to complex, industrially relevant CFD computations. These authors note that, in spite of the different nature of the Leonard stresses and cross-stresses, they are lumped together with the LES Reynolds stresses in the current versions of the finite volume method. The whole stress τ ij is modelled as a single entity by means of a single SGS turbulence

+ τ ii δ ij (3.94)

3 C ∂x

∂x i F 3

The Smagorinsky–Lilly SGS model builds on Prandtl’s mixing length model (3.39) and assumes that we can define a kinematic SGS viscosity ν

2 SGS

(dimensions m /s), which can be described in terms of one length scale and one velocity scale and is related to the dynamic SGS viscosity by ν SGS =

µ SGS / ρ. Since the size of the SGS eddies is determined by the details of the filtering function, the obvious choice for the length scale is the filter cutoff width ∆. As in the mixing length model, the velocity scale is expressed as

the product of the length scale ∆ and the average strain rate of the resolved flow ∆ × | D |, where | D | = 2D ij D ij . Thus, the SGS viscosity is evaluated as follows:

µ SGS = ρ(C SGS ∆) 2 |D|= ρ(C SGS ∆) 2 2 D ij D ij (3.95) where C SGS = constant

1 A ∂R ∂R D and D ij =

2 B C ∂x j ∂x E i F

Lilly (1966, 1967) presented a theoretical analysis of the decay rates of iso- tropic turbulent eddies in the inertial subrange of the energy spectrum, which suggests values of C SGS between 0.17 and 0.21. Rogallo and Moin (1984)

reviewed work by other authors suggesting values of C SGS = 0.19 – 0.24 for results across a range of grids and filter functions. They also quoted early LES computations by Deardorff (1970) of turbulent channel flow, which has strongly anisotropic turbulence, particularly in the near-wall regions. This work established that the above values caused excessive damping and suggested that C SGS = 0.1 is most appropriate for this type of internal flow calculation.

CHAPTER 3 TURBULENCE AND ITS MODELLING

The difference in C SGS values is attributable to the effect of the mean flow strain or shear. This gave an early indication that the behaviour of the small eddies is not as universal as was surmised at first and that successful LES turbulence modelling might require case-by-case adjustment of C SGS or a

more sophisticated approach.