Subharmonic eigenvalue orbits in the spectrum of pulsating Poiseuille flow

Abstract

Spectral degeneracies where eigenvalues and eigenvectors simultaneously coalesce, also known as exceptional points, are a natural consequence of the strong non-normality of the Orr–Sommerfeld operator describing the evolution of infinitesimal disturbances in parallel shear flows. While the resonances associated with these points give rise to algebraic growth, the development of non-modal stability theory exploiting specific perturbation structures with much larger potential for transient energy growth has led to waning interest in spectral degeneracies. The appearance of subharmonic eigenvalue orbits, recently discovered in the periodic spectrum of pulsating Poiseuille flow, can be traced back to the coalescence of eigenvalues at exceptional points. We present a thorough analysis of the spectral properties of the linear operator to identify exceptional points and accurately map the prevalence of subharmonic eigenvalue orbits for a large range of pulsation amplitudes and frequencies. This information is then combined with solutions of the linear initial value problem to analyse the impact of the appearance of these orbits on the temporal evolution of linear disturbances in pulsating Poiseuille flow. The periodic amplification phases are shown to be heralded by repeated non-normal growth bursts that are intensified by the formation of subharmonic orbits involving the leading eigenvalues. These bursts are associated with the change of alignment of the perturbation from the decaying towards the amplified branch of the subharmonic eigenvalue orbits in a so-called branch transition process.

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“…The chosen approach naturally captures linear phenomena that would appear under the quasi-steady assumption where it is appropriate. A detailed comparison of the two approaches is beyond the scope of this paper, but given the results obtained using the transient linear stability analysis framework presented here, the authors believe that an analysis similar to the one performed for pulsating Poiseuille flow (Kern et al 2021(Kern et al , 2022 would be an interesting extension of the present work to quantify the impact of the time dependence on the stability characteristics.…”

“…The chosen formulation using the specific rotation matrix (2.7) implies that adding more modes does not change existing basis vectors but allows a more accurate interpretation of the growth within the subspace in term of modal and non-modal growth, by adding more information about possible interactions with newly spanned directions that may be non-orthogonal (Kern et al 2022). Furthermore, tracking additional modes has the benefit of more accurately tracking the most unstable directions, which are effectively shielded from the dynamics outside of the subspace by the most stable modes (Babaee et al 2017).…”

“…There is indeed a risk for misinterpretation of non-modal growth with respect to the full operator as modal growth within the subspace that is unaware of non-orthogonal eigendirections of the full linear operator that it does not span. This topic is considered in more detail in Kern et al (2022) for the case of pulsatile plane Poiseuille flow. Especially, the modal growth rates recovered by a small OTD subspace should therefore not be considered accurate values, in particular in the context of the necessary spatial truncation of the subdomain that does not fully contain the modes.…”

“…The temporal periodicity of the flow case combined with its geometrical simplicity that admits an analytical solution even in the non-autonomous case – the direct comparison between the instantaneous eigendirections of the local operator (i.e. the quasi-steady analysis at each point in time) and the asymptotic linear solutions (the Floquet eigendirections) – was considered in Kern, Hanifi & Henningson (2022). One of the key results of the analysis was to show the increasing misalignment between the instantaneous eigendirections governed by the external time dependence via the baseflow and the perturbations, whose time scales are independent of the external pulsation time scale but are instead governed by the time scales of linear growth mechanisms, in this case the transient growth via the Orr mechanism in particular.…”

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“…Furthermore, as an exceptional point is a spectral singularity in parameter space, it limits the convergence of perturbation expansions (Orchini et al 2020a). When a periodic parameter variation is considered, such as in the spectrum of pulsating Poiseuille flow, the presence of an exceptional point may lead to subharmonic eigenvalue orbits (Kern et al 2022).…”