The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology

Abstract

The dimensionless dissipation rate constant Cϵ of homogeneous isotropic turbulence is such that Cϵ=f(logReλ)Cs′3, where f(logReλ) is a dimensionless function of logReλ which tends to 0.26 (by extrapolation) in the limit where logReλ⪢1 (as opposed to just Reλ⪢1) if the assumption is made that a finite such limit exists. The dimensionless number Cs′ reflects the number of large-scale eddies and is therefore nonuniversal. The nonuniversal asymptotic values of Cϵ stem, therefore, from its universal dependence on Cs′. The Reynolds number dependence of Cϵ at values of logReλ close to and not much larger than 1 is primarily governed by the slow growth (with Reynolds number) of the range of viscous scales of the turbulence. An eventual Reynolds number independence of Cϵ can be achieved, in principle, by an eventual balance between this slow growth and the increasing non-Gaussianity of the small scales. The turbulence is characterized by five length-scales in the following order of increasing magnitude: the Kolmogorov microscale η, the inner cutoff scale η*≈η(7.8+9.1logReλ), the Taylor microscale λ∼Reλ1∕2η, the voids length scale λv∼Reλ1∕3λ, and the integral length scale L∼Reλ2∕3λv.