Characterization of unsteady flow behavior by linear stability analysis

Abstract : Linear stability theory has been intensively used over the past decades for the characterization of unsteady flow behaviors. While the existing approaches are numerous, none has the ability to address any general flow. Moreover, clear validity conditions for these techniques are often missing, and this raises the question of their general reliability.In this thesis, this question is addressed by first considering the classical stability approach, which focuses on the evolution of small disturbances about a steady solution -- a base flow -- of the Navier-Stokes equations.To this end, the screech phenomenon -- a tonal noise that is sometimes generated by underexpanded jets -- is studied from alinear stability point of view. The results reveal that the nonlinear dynamics of this phenomenon is well-predicted by a linear base flow stability analysis. A confrontation with other similar analyses from the literature shows that such a satisfactory result is not always observed. However, when a self-sustained oscillating flow is driven by an acoustic feedback loop, as it is the case for the screech phenomenon, cavity flows and impinging jets for instance, then the nonlinearities have a weak impact on the frequency selection process, explaining the ability of a linear analysis to characterize the flow, even in the nonlinear regime.Another alternative approach, based on a linearization about the mean flow, is known to be successful in some cases where a base flow analysis fails. This observation from the literature is explained in this thesis by outlining the role of the resolvent operator, arising from a linearization about the mean flow, in the dynamics of a flow. The main finding is that if this operator displays a clear separation of singular values, which relates to the existence of one strong convective instability mechanism, then the Fourier modes areproportional to the first resolvent modes. This result provides mathematical and physical conditions for the use and meaning of several mean flow stability techniques, such as a parabolised stability equations analysis of a mean flow.Moreover, it leads to a predictive model for the frequency spectrum of a flow field at any arbitrary location, from the sole knowledge of the mean flow and the frequency spectrum at one or more points. All these findings are illustrated and validated in the case of a turbulent backward facing step flow. Finally, these results are exploited in an experimental context, for the reconstruction of the unsteady behavior of a transitional round jet, from the sole knowledge of the mean flow and one point-wise measurement. The study shows that, after following a few experimental precautions, detailed in the manuscript, the reconstruction is very accurate and robust.
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Samir Beneddine. Characterization of unsteady flow behavior by linear stability analysis. Mechanics of the fluids [physics.class-ph]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLX010⟩. ⟨tel-01513332⟩

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