  We establish the incompressible Navier--Stokes limit for the discrete
velocity model of the Boltzmann equation in any dimension of the physical
space, for densities which remain in a suitable small neighborhood of the
global Maxwellian.
  Appropriately scaled families solutions of discrete Boltzmann equation are
shown to have fluctuations that locally in time converge strongly to a limit
governed by a solution of Incompressible Navier--Stokes provided that the
initial fluctuation is smooth, and converges to appropriate initial data.
  As applications of our results, we study the Carleman model and the
one-dimensional Broadwell model.
