On the multifractal properties of the energy dissipation derived from turbulence data

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Abstract
Various difficulties can be expected in trying to extract from experimental data the distribution of singularities, the f(α) function, of the energy dissipation. One reason is that the multifractal model of turbulence implies a dependence of the viscous cutoff on the singularity exponent. It is an open question if current hot-wire probes can resolve the scales implied by exponents a significantly less than 1.Two exactly soluble models are used to show how spurious scaling can occur, due to finite Reynolds number effects. In the Gaussian model the true velocity signal is replaced by independent Gaussian random variables. The dissipation, defined as the square of the difference of successive variables, has trivial scaling in so far as all the moments of spatial averages of the dissipation behave asymptotically as a uniform dissipation. Still, contamination by subdominant terms requires that scaling exponents for high-order moments be identified over an increasingly large range of scales. If the available range is limited by the Reynolds number, scaling exponents for high orders will be systematically underestimated and spurious intermittency will be inferred. Burgers’ model is used to highlight further problems. At finite Reynolds numbers, regions with no small-scale activity (away from shocks) have a residual dissipation which contributes a spurious point (α = 1,f(α) = 1). In addition, when the f(α) function is obtained by Legendre transform techniques, convex hull effects generate an entire spurious segment.Finally, Burgers’ model also indicates that the relation between exponents of structure functions and exponents of local dissipation moments, which goes back to Kolmogorov's (1962) work, leads to an inconsistency for structure functions of low positive order.