High-order structure functions of turbulence

Abstract

The scaling behaviour of high-order structure functions G_p(r) =〈(u(x+r)−u(x))^p〉 is studied in a variety of laboratory turbulent flows. The statistical accuracy of the structure function benefits from novel instrumentation for its real-time measurement. The nature of statistical errors is discussed extensively. It is argued that integration times must increase for decreasing separations r. Based on the statistical properties of probability density functions we derive a simple estimate of the required integration time for moments of a given order. We further give a way for improving this accuracy through careful extrapolation of probability density functions of velocity differences.

Structure functions are studied in two different kinematical situations. The (standard) longitudinal structure functions are measured using Taylor's hypothesis. In the transverse case an array of probes is used and no recourse to Taylor's hypothesis is needed. The measured scaling exponents deviate from Kolmogorov's (1941) prediction, more strongly so for the transverse exponents.

The experimental results are discussed in the light of the multifractal model that explains intermittency in a geometrical framework. We discuss a prediction of this model for the form of the structure function at scales where viscosity becomes of importance.