Fractal properties of inertial-range turbulence with implications for aircraft response

Abstract

Summary Fractal geometry provides a method for modelling the scale dependence of fluctuations in atmospheric-turbulence velocity. In this paper the basic concepts are outlined and illustrated by a method of data analysis which, for a fractal process, displays measured probability distributions in scale-invariant form. To a first approximation the data exhibit statistical self-similarity, consistent with the classical theory of Kolmogorov. However, on more detailed analysis, the more intense fluctuations show systematic departures from self-similarity, consistent with recent theoretical estimates of the fractal dimension of the support of turbulence. Implications for aircraft gust response are discussed.

Wavelet analysis of gust structure in measured atmospheric turbulence data

“…These results have important implications 13 for the prediction of structural loads and the performance of load-alleviation control systems, for each of which gusts with scale L less than 100 m can be critical. The correct representation in aircraft design criteria of short gusts is of particular relevance to modern technology aircraft which use powerful active-control systems, as the effect of such systems is sometimes to reduce the scale L of the gusts to which the aircraft predominantly responds.…”

“…13 were the results of analyzing a particular sample of turbulence from these flight measurements using the self-similar model (D = 1, k = i) and also using the more general (mono)fractal model with the indices D and k treated as independent free parameters. Plots of S D n(L, z) against zlL k showed that for this sample an improved fit of the model to the data was achieved at the larger amplitudes with empirically-fitted exponents D = 0.45 and k = 0.23.…”

Parametric modelling of turbulence

“…The developing theory of chaos and of fractals has, in recent years, thrown interesting light on aspects of turbulence (see, for example, Mandelbrot 1976Mandelbrot , 1982Eckmann & Ruelle 1985;Jones et al 1988), whereas there has been relatively little progress in the study of turbulence by the statistical approach. Most of the existing statistical treatm ents are based on second-and higher-order moment properties or quite general studies of questions concerning stochastic solutions of the Navier-Stokes equations, or both (see, for example, Monin & Yaglom 1975;Vishik et al 1979;Tatsumi et al 1986), rather than on integrated, parametric modelling by means of stochastic process theory.…”