On the deficiency of even-order structure functions as inertial-range diagnostics

Abstract

In the limit of vanishing viscosity, ν→0, Kolmogorov's two-thirds, 〈(Δυ)²〉~ε^2/3r^2/3, and five-thirds, E~ε^2/3k^−5/3, laws are formally equivalent. (Here 〈(Δυ)²〉 is the second-order structure function, ε the dissipation rate, r the separation in physical space, E the three-dimensional energy spectrum, and k the wavenumber.) However, for the Reynolds numbers encountered in terrestrial experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. We ask why this should be. To this end, we create artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled. We choose the energy of eddies of scale, s, to vary as s^2/3, in accordance with Kolmogorov's 1941 law, and vary the range of scales, γ=s_max/s_min, in any one realization from γ=25 to γ=800. This is equivalent to varying the Reynolds number in an experiment from R_λ=60 to R_λ=600. We find that, while there is some evidence of a five-thirds law for γ>50; (R_λ>100), the two-thirds law only starts to become apparent when γ approaches 200 (R_λ~240). The reason for this discrepancy is that the second-order structure function is a poor filter, mixing information about energy and enstrophy, and from scales larger and smaller than r. In particular, in the inertial range, 〈(Δυ)²〉 takes the form of a mixed power law, a₁ + a₂r² + a₃r^2/3, where a₂r² tracks the variation in enstrophy and a₃r^2/3 the variation in energy. These findings are shown to be consistent with experimental data where the ‘pollution’ of the r^2/3 law by the enstrophy contribution, a₂r², is clearly evident. We show that higher-order structure functions (of even order) suffer from a similar deficiency.