  This work aims to numerically investigate the linear global instability of
long separation bubbles origin from the changes in the adverse pressure
gradient inside a laminar flat plate boundary layer disturbed by placing a
bluff body with small clearances. The boundary layer starts at
$Re_{\delta^\ast}$=121.7 and is steady in the computational extent after being
disturbed by a NACA 4415 airfoil and a cylinder with clearance $h\leqslant
0.2$, respectively. It is found that the dominant stability is associated with
two- and three-dimensional stationary eigenmodes. There is a less damped
oscillation mode at a small wavenumbers range. The strong effect of
three-dimensionality is further confirmed in a non-modal analysis framework.
Transient growth analysis shows that the most strongest optimal perturbation is
three-dimensional in the range of parameters studied. The modal analysis
reveals that the instability grows with reducing the clearance, while the
optimal perturbations grow with increasing the clearance. It is found that the
separated boundary layer stability analysis is less dependent on the bluff body
geometries.
