Turbulent cascades: Limitations and a statistical test of the lognormal hypothesis

Abstract

Multiplicative random cascade models were introduced in the 1970s to explain the intermittency of turbulent energy dissipation. The rigorous results in the multifractality of cascade measures recently derived by this author are used in two ways. (1) The statistical test for the Kolmogorov–Obukhov lognormal hypothesis (K62) is revised. Contrary to what is generally believed, we show that the K62 theoretical prediction is in good agreement with experimental data in the range (1,18) of the parameter p (the order of velocity structure functions). This revised conclusion was necessitated by violations in previous comparisons of the “ergodic hypothesis” for large p. (2) Physical limitations on cascade models are analyzed. We show that cascade measures demonstrate a strict dependence on the scaling parameter. This circumstance affects interpretations of statistics of multipliers, shows that the models used in practice are not really superior to others, and indicates the necessity to study cascades with a random scaling parameter.