In a recent paper we presented evidence for the occurence of Leray-like singularities with positive Sedov-Taylor exponent $alpha$ in turbulent flows recorded in Modanes wind tunnel, by looking at simultaneous acceleration and velocity records. Here we use another tool which allows to get other informations on the dynamics of turbulent bursts. We compare the structure functions for velocity and acceleration in the same turbulent flows. This shows the possible contribution of other types of self-similar solutions because this new study shows that statistics is seemingly dominated by singularities with small positive or even negative values of the exponent $alpha$, that corresponds to weakly singular solutions with singular acceleration, and regular velocity. We present several reasons explaining that the exponent $alpha$ derived from the structure functions curves, may look to be negative.
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