  We investigate the statistical properties, based on numerical simulations and
analytical calculations, of a recently proposed stochastic model for the
velocity field of an incompressible, homogeneous, isotropic and fully developed
turbulent flow. A key step in the construction of this model is the
introduction of some aspects of the vorticity stretching mechanism that governs
the dynamics of fluid particles along their trajectory. An additional further
phenomenological step aimed at including the long range correlated nature of
turbulence makes this model depending on a single free parameter $\gamma$ that
can be estimated from experimental measurements. We confirm the realism of the
model regarding the geometry of the velocity gradient tensor, the power-law
behaviour of the moments of velocity increments (i.e. the structure functions),
including the intermittent corrections, and the existence of energy transfers
across scales. We quantify the dependence of these basic properties of
turbulent flows on the free parameter $\gamma$ and derive analytically the
spectrum of exponents of the structure functions in a simplified non
dissipative case. A perturbative expansion in power of $\gamma$ shows that
energy transfers, at leading order, indeed take place, justifying the
dissipative nature of this random field.
