  The stability of a three-dimensional, incompressible, viscous flow through a
finite-length duct is studied. A divergence-free basis technique is used to
formulate the weak form of the problem. A SUPG (streamingline upwind
Petrov-Galerkin) based scheme for eigenvalue problems is proposed to stabilize
the solution. With proper boundary condtions, the least-stable eigenmodes and
decay rates are computed. It is again found that the flows are asymptotically
stable for all $Re$ up to $2500$. It is discovered that the least-stable
eigenmodes have a boundary-layer-structure at high $Re$, although the
Poiseuille base flow does not exhibits such structure. At these Reynolds
numbers, the eigenmodes are dominant in the vicinity of the duct wall and are
convected downstream. The boundary-layer-structure brings singularity to the
modes at high $Re$ with unbounded perturbation gradient. It is shown that due
to the singular structure of the least-stable eigenmodes, the linear
Navier-Stoker operator tends to have pseudospectrua and the nonlinear mechanism
kicks in when the perturbation energy is still small at high $Re$. The
decreasing stable region as $Re$ increases is a result of both the decreasing
decay rate and the singular structure of the least-stable modes. The result
demonstrated that at very high $Re$, linearization of Navier-Stokes equation
for duct flow may not be a good model problem with physical disturbances.
