  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 Modane's 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.
