  We consider the Navier--Stokes equations in a half-plane with a drift term
parallel to the boundary and a small source term of compact support. We provide
detailed information on the behavior of the velocity and the vorticity at
infinity in terms of an asymptotic expansion at large distances from the
boundary. The expansion is universal in the sense that it only depends on the
source term through some multiplicative constants. This expansion is identical
to the one for the problem of an exterior flow around a small body moving at
constant velocity parallel to the boundary, and can be used as an artificial
boundary condition on the edges of truncated domains for numerical simulations.
