Determination of the Smagorinsky–Lilly constant CS

The Smagorinsky–Lilly (SL) SGS model ${\nu }_t{\mathrm{=(C}}_S{\mathrm{\Delta )}}^{2}S$ yields a constant $C_S\text{=0.20–0.22}$ which is a factor of 2 larger than what is needed in LES calculations; in addition, Deardorff and Hunt et al. suggested empirical corrections to the SL model to account for the effects of stratification and shear. In this paper, we propose an SGS model that naturally includes stratification and shear (recovering the two previous models) and that gives rise to a value of $C_S\text{∼0.11.}$ The three basic assumptions underlying the SL model are (1) Fickian approximation, ${b}{\text{=−2ν}}_t{S},$ where b is the Reynolds stress tensor and S is the strain rate tensor, (2) SGS satisfy Kolmogorov law, and (3) local equilibrium, $\mathrm{P=\epsilon ,}$ where $P$ and ε are the rates of production and dissipation. We avoid (1) by using the most general ${b}={b}({S},{R})$ relationship, where R is the vorticity, and (3) by letting the ratio $\mathrm{P/\epsilon }$ vary. The most critical ingredient is (2). We derive the energy spectrum $\mathrm{E(k)}$ in the presence of buoyancy $N$ and shear $S$ and show that the SGS scales are not Kolmogorov, which sets in only for wave numbers $\mathrm{k\gg \pi /\Delta .}$ Integrating over all SGS scales we obtain the turbulent kinetic energy and then construct a new dissipation length $\mathrm{l=l(N,S),}$ which we validate in three ways: (a) use of $\text{l(N,0)}$ reproduces the empirical SGS model by Deardorff, (b) use of $\text{l(0,S)}$ reproduces the empirical SGS model of Hunt et al., and (c) the complete $\mathrm{l(N,S)}$ reproduces recent LES data that no other has been able to explain. Results for $C_S$ are as follows. Homogeneous shear: the removal of each of the three approximations is responsible for an almost equal (∼30%) lowering of $C_S$ from 0.2 to 0.1. Plane strain: the lack of vorticity makes the Fickian approximation an acceptable one. The lowering of $C_S$ is due in equal measure to the removal of (2) and (3) above. The two examples show that even though the numerical value of ∼0.11 may look like a “universal” constant, it is actually the combination of physical processes that differ from flow to flow. That $C_S$ is actually a dynamical variable that adjusts itself to each flow has already been observed by previous authors.