Modal and nonmodal stability of a stably stratified boundary layer flow
Résumé
The modal and nonmodal linear stability of a stably stratified Blasius boundary layer
flow, composed of a velocity and a thermal boundary layer, is investigated. The temporal
and spatial linear stability of such flow is investigated for several Richardson, Reynolds,
and Prandtl numbers. While increasing the Richardson number stabilizes the flow, a more
complex behavior is found when changing the Prandtl number, leading to a stabilization
of the flow up to Pr = 7, followed by a destabilization. The nonmodal linear stability
of the same flow is then investigated using a direct-adjoint procedure optimizing four
different approximations of the energy norm based on a weighted sum of the kinetic
and the potential energies. No matter the norm approximation, for short target times an
increase of the Richardson number induces a decrease of the optimal energy gain and
time at which it is obtained and an increase of the optimal streamwise wave number,
which considerably departs from zero. Moreover, the dependence of the energy growth
on the Reynolds number transitions from quadratic to linear, whereas the optimal time,
which varies linearly with Re in the nonstratified case, remains constant. This suggests
that the optimal energy growth mechanism arises from the joint effect of the lift-up and
the Orr mechanism, that simultaneously act to increase the shear production term on a
rather short timescale, counterbalancing the stabilizing effect of the buoyancy production
term. Although these short-time mechanisms are found to be robust with respect to the
chosen norm, a different amplification mechanism is observed for long target times for
three of the proposed norms. This strong energy growth, due to the coupling between
velocity and temperature perturbations in the free stream, disappears when the variation
of the stratification strength with height is accurately taken into account in the definition of
the norm.
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