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 $\text{Pr}=7$, 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.

中文翻译：

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稳定分层边界层流的模态和非模态稳定性

研究了由速度和热边界层组成的稳定分层的Blasius边界层流的模态和非模态线性稳定性。对于几个理查森数，雷诺数和普朗特数，研究了这种流动的时间和空间线性稳定性。虽然增加Richardson数可稳定流量，但更改Prandtl数时会发现更复杂的行为，从而导致流量稳定到最大。$\text{镨}=7$，然后造成不稳定。然后使用直接伴随过程研究相同流的非模态线性稳定性，该过程基于动能和势能的加权总和优化了能量范数的四个不同近似值。不管范数近似如何，对于较短的目标时间，理查森数的增加都会导致最佳能量增益和获得该能量的时间减少，并且最佳流向波数也会增加，而零值会大大偏离。此外，能量增长对雷诺数从二次到线性转变的依赖性，而在非分层情况下，最佳时间随Re线性变化。这表明最佳的能量增长机制来自提举和Orr机制的共同作用，同时可以在很短的时间内增加剪力生产期，从而抵消了浮力生产期的稳定作用。尽管发现这些短时机制相对于所选规范具有鲁棒性，但对于三个拟议规范，对于较长的目标时间却观察到不同的放大机制。当在规范的定义中准确考虑分层强度随高度的变化时，由于自由流中速度和温度扰动之间的耦合而导致的这种强大的能量增长就消失了。尽管发现这些短时机制相对于所选规范具有鲁棒性，但对于三个拟议规范，对于较长的目标时间却观察到不同的放大机制。当在规范的定义中准确考虑分层强度随高度的变化时，由于自由流中速度和温度扰动之间的耦合而导致的这种强大的能量增长就消失了。尽管发现这些短时机制相对于所选规范具有鲁棒性，但对于三个拟议规范，对于较长的目标时间却观察到不同的放大机制。当在规范的定义中准确考虑分层强度随高度的变化时，由于自由流中速度和温度扰动之间的耦合而导致的这种强大的能量增长就消失了。