  We study the three-dimensional turbulent Kolmogorov flow, i.e. the
Navier-Stokes equations forced by a low-single-wave-number sinusoidal force in
a periodic domain, by means of direct numerical simulations. This classical
model system is a realization of anisotropic and non-homogeneous hydrodynamic
turbulence. Boussinesq's eddy viscosity linear relation is checked and found to
be approximately valid over half of the system volume. A more general nonlinear
quadratic constitutive equation is proposed and its parameters estimated at
varying the Taylor scale-based Reynolds number in the flow up to the value 200.
This provides a Reynolds number-dependent quadratic closure for the Kolmogorov
flow. The case of a forcing with a different shape, here chosen Gaussian, is
considered and the differences with the sinusoidal forcing are emphasized.
