  A coarse-grained particle model for incompressible Navier-Stokes (NS)
equation is proposed based on spatial filtering by utilizing smoothed particle
hydrodynamics (SPH) approximations. This model is similar to our previous
developed SPH discretization of NS equation ({\it Hu X.Y. & N.A. Adams, J.
Comput. Physics}, 227: 264-278, 2007 and 228: 2082-2091, 2009) and the
Lagrangian averaged NS (LANS-$\alpha$) turbulence model. Other than using
smoothing approaches, this model obtains particle transport velocity by
imposing constant $\sigma$ which is associated with the particle density, and
is called SPH-$\sigma$ model. Numerical tests on two-dimensional decay and
forced turbulences with high Reynolds number suggest that the model is able to
reproduce both the inverse energy cascade and direct enstrophy cascade of the
kinetic energy spectrum, the time scaling of enstrophy decay and the
non-Guassian probability density function (PDF) of particle acceleration.
