  In this paper, we prove the uniform nonlinear structural stability of
Poiseuille flows with suitably large flux for the steady Navier-Stokes system
in a two-dimensional strip with arbitrary period. Furthermore, the
well-posedness theory for the Navier-Stokes system is also proved even when the
$L^2$-norm of the external force is large. In particular, if the vertical
velocity is suitably small where the smallness is independent of the flux, then
Poiseuille flow is the unique solution of the steady Navier-Stokes system in
the periodic strip. The key point is to establish uniform a priori estimates
for the corresponding linearized problem via the boundary layer analysis, where
we explore the particular features of odd and even stream functions. The
analysis for the even stream function is new, which not only generalizes the
previous study for the symmetric flows in \cite{Rabier1}, but also provides an
explicit relation between the flux and period.
