We experimentally study the properties of mean and most probable velocity fields in a turbulent von K{á}rm{á}n 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.