  We consider here the Navier-Stokes equations in $\mathbb{R}^{3}$ with a
stationary, divergence-free external force and with an additional damping term
that depends on two parameters. We first study the well-posedness of weak
solutions for these equations and then, for a particular set of the damping
parameters, we will obtain an upper and lower control for the energy
dissipation rate $\varepsilon$ according to the Kolmogorov K41 theory. However,
although the behavior of weak solutions corresponds to the K41 theory, we will
show that in some specific cases the damping term introduced in the
Navier-Stokes equations could annihilate the turbulence even though the Grashof
number (which are equivalent to the Reynolds number) are large.
