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In the study of turbulence, intermittency is a measure of how muchKolmogorov's theory of 1941 deviates from experiments. It is quantified withthe flatness of the velocity of the fluid, usually based on structure functionsin the physical space. However, it can also be defined with Fourier high-passfilters. Experimental and numerical simulations suggest that the two approachesdo not always give the same results. Our purpose is to compare them from theanalytical point of view of functions. We do that by studying generalizationsof Riemann's non-differentiable function, yielding computations that arerelated to some classical problems in Fourier analysis. The conclusion is thatthe result strongly depends on regularity. To visualize this, we establish ananalogy between these generalizations and the influence of viscosity inturbulent flows. This article is motivated by the mathematical works on themultifractal formalism and the discovery of Riemann's non-differentiablefunction as a trajectory of polygonal vortex filaments.

An analytical study of flatness and intermittency through Riemann's non-differentiable functions