  We study a colocated cell centered finite volume method for the approximation
of the incompressible Navier-Stokes equations posed on a 2D or 3D finite
domain. The discrete unknowns are the components of the velocity and the
pressures, all of them colocated at the center of the cells of a unique mesh;
hence the need for a stabilization technique, which we choose of the
Brezzi-Pitk\"aranta type. The scheme features two essential properties: the
discrete gradient is the transposed of the divergence terms and the discrete
trilinear form associated to nonlinear advective terms vanishes on discrete
divergence free velocity fields. As a consequence, the scheme is proved to be
unconditionally stable and convergent for the Stokes problem, the steady and
the transient Navier-Stokes equations. In this latter case, for a given
sequence of approximate solutions computed on meshes the size of which tends to
zero, we prove, up to a subsequence, the $L^2$-convergence of the components of
the velocity, and, in the steady case, the weak $L^2$-convergence of the
pressure. The proof relies on the study of space and time translates of
approximate solutions, which allows the application of Kolmogorov's theorem.
The limit of this subsequence is then shown to be a weak solution of the
Navier-Stokes equations. Numerical examples are performed to obtain numerical
convergence rates in both the linear and the nonlinear case.
