  We introduce a continuous data assimilation (downscaling) algorithm for the
two-dimensional Navier-Stokes equations employing coarse mesh measurements of
only one component of the velocity field. This algorithm can be implemented
with a variety of finitely many observables: low Fourier modes, nodal values,
finite volume averages, or finite elements. We provide conditions on the
spatial resolution of the observed data, under the assumption that the observed
data is free of noise, which are sufficient to show that the solution of the
algorithm approaches, at an exponential rate asymptotically in time, to the
unique exact unknown reference solution, of the 2D Navier-Stokes equations,
associated with the observed (finite dimensional projection of) velocity.
