School of Engineering, The University of Queensland
Abstract
Kolmogorov’s two-thirds, h(Dv)2i e2/3r2/3, and five-thirds, E e2/3k−5/3, laws are formally equivalent in the limit of vanishing viscosity, n!0. However, for most Reynolds numbers encountered in laboratory scale experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. By creating artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled, we show why this is the case. The energy of eddies of scale, s, is shown to vary as s2/3, in accordance with Kolmogorov’s 1941 law, and we vary the range of scales, g = smax/smin, in any one realisation from g = 25 to g = 800. This is equivalent to varying the Reynolds number in an experiment from Rl = 60 to Rl = 600. While there is some evidence of a five-thirds law for g > 50 (Rl > 100), the two-thirds law only starts to become apparent when g approaches 200 (Rl 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, h(Dv)2i takes the form of a mixed power-law, a1+a2r2+a3r2/3, where a2r2 tracks the variation in enstrophy and a3r2/3 the variation in energy. These findings are shown to be consistent with experimental data where the polution of the r2/3 law by the enstrophy contribution, a2r2, is clearly evident. We show that higherorder structure functions (of even order) suffer from a similar deficiency. (See also [2].
