Intermittent turbulence and fractal dimension: kurtosis and the spectral exponent 5/3 + B

Various distinct aspects of the geometry of turbulence can be studied with the help of a wide family of shapes for which I have recently coined the neologism “fractals.” These shapes are loosely characterized as being violently convoluted and broken up, a feature denoted in Latin by the adjective “fractus,” Fractal geometry approaches the loose notion of “form” in a manner different and almost wholly separate from the approach used by topology. Until recently, it was believed that fractals could not be of use in concrete applications, but I have shown them to be useful in a variety of fields. In particular, they play a central role in the study of three aspects of turbulence: (a) homogeneous turbulence, through the shape of the iso-surfaces of scalars (M 1975f{H}), (b) turbulent dispersion (M 1976c{N20}), and especially c) the intermittency of turbulent dissipation (M 1972j {N14}, M1974f{N15} and M1974c{N16}. The present paper will sketch a number of links between the new concern with fractal geometry and the traditional concerns with various spectra of turbulence and the kurtosis of dissipation. Some of the results to be presented will improve and/or correct results found in the literature, for example, will further refine M 1974f,{N15}, and M1974c{N16}. One result described in Section 4 deserves special emphasis: It confirms that one effect of the intermittency of dissipation is to replace the classical spectral exponent 5/3 by 5/3 + B. However, it turns out that in the general case the value of B is different from the value accepted in the literature, for example in Monin & Yaglom 1975. The accepted value, derived by Kolmogorov, Obukhov, and Yaglom, is linked to a separate Ansatz, called lognormal hypothesis, and is shown in M 1974f{N15} to be highly questionable. Other results in this paper are harder to state precisely in a few words. They demonstrate the convenience and heuristic usefulness of the fractally homogeneous approximation to intermittency, originating in Berger & M 1963{N6} and Novikov & Stewart 1964. They also demonstrate the awkwardness of the lognormal hypothesis. Many writers of the Russian school have noted that the latter is only an approximation, but it becomes increasingly clear that even they underestimated its propensity to generate paradoxes and to hide complexities. M 1975O describes numerous other concrete applications of fractals. It can also serve as a general background reference, but its chapter on turbulence is too skimpy to be of use here. This deficiency should soon be corrected in the English version, M 1977F, meant to serve as a preface to technical works such as the present one. Nevertheless, in its main points, the present text is self-contained.