  We perform resolvent analysis to examine the perturbation dynamics over the
laminar separation bubble (LSB) that forms near the leading edge of a NACA 0012
airfoil at a chord-based Reynolds number of 500,000 and an angle of attack of 8
degrees. Randomized SVD is adopted in the present analysis to relieve the
computational cost associated with the high-Re global base flow. To examine the
local physics over the LSB, we consider the use of exponential discounting to
limit the time horizon that allows for the instability to develop with respect
to the base flow. With discounting, the gain distribution over frequency
accurately captures the spectral content over the LSB obtained from flow
simulation. The peak-gain frequency also agrees with previous flow control
results on suppressing dynamic stall over a pitching airfoil. According to the
gain distribution and the modal structures, we conclude that the dominant
energy-amplification mechanism is the Kelvin-Helmholtz instability. In addition
to discounting, we also examine the use of spatial windows for both the forcing
and response. From the response-windowed analysis, we find that the LSB serves
the main role of energy amplifier, with the amplification saturating at the
reattachment point. The input window imposes the constraint of surface forcing,
and the results show that the optimal actuator location is slightly upstream of
the separation point. The surface-forcing mode also suggest the optimal
momentum forcing in the surface-tangent direction, with strong
uni-directionality that is ideal for synthetic-jet-type actuators. This study
demonstrates the strength of randomized resolvent analysis in tackling
high-Reynolds-number base flows and calls attention to the care needed for
base-flow instabilities.
