  A phenomenological theory of the fluctuations of velocity occurring in a
fully developed homogeneous and isotropic turbulent flow is presented. The
focus is made on the fluctuations of the spatial (Eulerian) and temporal
(Lagrangian) velocity increments. The universal nature of the intermittency
phenomenon as observed in experimental measurements and numerical simulations
is shown to be fully taken into account by the multiscale picture proposed by
the multifractal formalism, and its extensions to the dissipative scales and to
the Lagrangian framework. The article is devoted to the presentation of these
arguments and to their comparisons against empirical data. In particular,
explicit predictions of the statistics, such as probability density functions
and high order moments, of the velocity gradients and acceleration are derived.
In the Eulerian framework, at a given Reynolds number, they are shown to depend
on a single parameter function called the singularity spectrum and to a
universal constant governing the transition between the inertial and
dissipative ranges. The Lagrangian singularity spectrum compares well with its
Eulerian counterpart by a transformation based on incompressibility,
homogeneity and isotropy and the remaining constant is shown to be difficult to
estimate on empirical data. It is finally underlined the limitations of the
increment to quantify accurately the singular nature of Lagrangian velocity.
This is confirmed using higher order increments unbiased by the presence of
linear trends, as they are observed on velocity along a trajectory.
