Breakdown of Kolmogorov's first similarity hypothesis in grid turbulence

Abstract

Kolmogorov's first similarity hypothesis (or KSH1) stipulates that two-point statistics have a universal form which depends on two parameters, the kinematic viscosity ν and the mean energy dissipation rate ⟨ε⟩. KSH1 is underpinned by two assumptions: the Reynolds number is very large and local isotropy holds. To disentangle the intricacies of these two requirements, we assess the validity of KSH1 in a flow where local isotropy is a priori tenable, i.e. decaying grid turbulence. The main question we address is how large should the Reynolds number be for KSH1 to be valid over a range of scales wider than, say, five Kolmogorov scales. To this end, direct numerical simulations based on the lattice Boltzmann method are carried out in low Reynolds number grid turbulence. The results show that when the Taylor microscale Reynolds number Rλ drops below about 20, the Kolmogorov normalised spectra deviate from those at higher Rλ; the deviation increases with decreasing Rλ. It is shown that at Rλ ≃ 20, the contribution of the energy transfer in the scale-by-scale energy budget becomes smaller than the contributions from the viscous and (large-scale) non-homogeneous terms at all scales, but never vanishes, at least for the range of Reynolds investigated here. A phenomenological argument based on the ratio N between the energy-containing timescale and the dissipative range timescale leads to the condition for KSH1 to hold. The numerical data indicate that N = 5, yielding Rλ ≃ 20, thus confirming our numerical finding. The present results show that KSH1, unlike the second Kolmogorov similarity hypothesis (KSH2,) does not require the existence of an inertial range. While it may seem remarkable that KSH1 is validated at much lower Reynolds numbers than required for KSH2 in grid turbulence (Rλ ≥ 1000,), KSH1 applies to small scales which include both dissipative scales and inertial range (if it exists). One can expect that, as the Reynolds number increases, the dissipative scales should satisfy KSH1 first; then, as the Reynolds number attains very high values, the inertial range is established in conformity with KSH2.