Modal and nonmodal stability of a stably stratified boundary layer flow

Abstract

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.