  We experimentally study the properties of mean and most probable velocity
fields in a turbulent von K\'arm\'an flow. These fields are found to be
described by two families of functions, as predicted by a recent statistical
mechanics study of 3D axisymmetric flows. We show that these functions depend
on the viscosity and on the forcing. Furthermore, when the Reynolds number is
increased, we exhibit a tendency for Beltramization of the flow, i.e. a
velocity-vorticity alignment. This result provides a first experimental
evidence of nonlinearity depletion in non-homogeneous non-isotropic turbulent
flow.
