A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type LES

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Abstract
A Kolmogorov-type similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type large-eddy simulation (LES) at very large LES Reynolds numbers is developed and discussed. The underlying concept is that the LES equations may be considered equations of motion of specific hypothetical fully turbulent non-Newtonian fluids, called ‘LES fluids’. It is shown that the length scale ls = csδ, which scales the magnitude of the variable viscosity in a Smagorinsky-type LES, is the ‘Smagorinsky-fluid’ counterpart of Kolmogorov's dissipation length for a Newtonian fluid where ν is the kinematic viscosity and ε is the energy dissipation rate. While in a Newtonian fluid the viscosity is a material parameter and the length ν depends on ε, in a Smagorinsky fluid the length ls is a material parameter and the viscosity depends on ε. The Smagorinsky coefficient cs may be considered the reciprocal of a ‘microstructure Knudsen number’ of a Smagorinsky fluid. A combination of Lilly's (1967) cut-off model with two well-known spectral models for dissipation-range turbulence (Heisenberg 1948; Pao 1965) leads to models for the LES-generated Kolmogorov coefficient αLES as a function of cs. Both models predict an intrinsic overestimation of αLES for finite values of cs. For cs = 0.2 Heisenberg's and Pao's models provide αLES = 1.74 (16% overestimation) and αLES = 2.14 (43% overestimation), respectively, if limcs → (αLES) = 1.5 is ad hoc assumed. The predicted overestimation becomes negligible beyond about cs = 0.5. The requirement cs > 0.5 is equivalent to Δ < 2ls. A similar requirement, L < 2η where L is the wire length of hot-wire anemometers, has been recommended by experimentalists. The value of limcs → (αLES) for a Smagorinsky-type LES at very large LES Reynolds numbers is not predicted by the models and remains unknown. Two critical values of cs are identified. The first critical cs is Lilly's (1967) value, which indicates the cs below which finite-difference-approximation errors become important; the second critical cs is the value beyond which the Reynolds number similarity is violated.