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

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 scalels=csδ, which scales the magnitude of the variable viscosity in a Smagorinsky-type LES, is the ‘Smagorinsky-fluid’ counterpart of Kolmogorov's dissipation length$\eta = v^{3/4}\epsilon^{-1/4}$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 lengthlsis a material parameter and the viscosity depends on ε. The Smagorinsky coefficientcsmay 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 αLESas a function ofcs. Both models predict an intrinsic overestimation of αLESfor finite values ofcs. Forcs= 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 isad hocassumed. The predicted overestimation becomes negligible beyond aboutcs= 0.5. The requirementcs> 0.5 is equivalent to Δ < 2ls. A similar requirement,L< 2η whereLis 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 ofcsare identified. The first criticalcsis Lilly's (1967) value, which indicates thecsbelow which finite-difference-approximation errors become important; the second criticalcsis the value beyond which the Reynolds number similarity is violated.