In numerical simulations with highly turbulent flows, the smallest scales are filtered; thereby, the effects of intermittency on those scales are neglected. When the flow is loaded by heavy small particles, the decimation of rapid changes in the velocity may lead to wrong results. This paper provides an approach to take account of subfiltered events of strong velocity jumps on the motion of heavy small particles. The idea is to force the filtered Navier-Stokes equations by a stochastic acceleration term with statistical properties identified by experiments and DNS. To this end, the stochastic model for supplemented acceleration contains the lognormal stochastic process for its norm (with long-range correlations) and the new stochastic model for the acceleration direction (with short-range correlations). The latter represents the Ornstein-Uhlenbeck process in Cartesian coordinates with relaxation to the local direction of the resolved vorticity; thereby, the geometry of highly stretched vortical structures is introduced in the designed model. Both stochastic processes depend on the local Reynolds number. The proposed flow model is applied for simulation of the background turbulence in which heavy particles are released and tracked. The assessment of single and two-time statistics of the particle acceleration and velocity clearly illustrates the advantage of the proposed flow model.