Abstract
Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr–Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien–Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.
Highlights
Since the seminal work of Reynolds, to whom we owe the ubiquitous Reynolds number, on transition to turbulence in pipes (Reynolds 1883) and the discovery of the famed inflection 927 A6-1J.S
We introduce the flow case used to illustrate and validate the implementation in Nek5000 (§ 4) and present the results of the transient linear stability analysis based on the optimally time-dependent (OTD) modes (§ 6)
The Cauchy–Green strain tensor is symmetric positive definite and has an orthogonal eigenbasis ξi with positive eigenvalues μi. While it follows from the equivalence between Gram–Schmidt vectors and the OTD basis vectors that the OTD modes converge to the dominant eigendirections of the asymptotic Cauchy–Green tensor, it was independently shown by Babaee et al (2017) that this convergence is exponentially fast and that it is possible to compute the finite-time Lyapunov exponents (FTLEs) as a byproduct of the computation of the OTD basis at negligible extra cost, independently of the dimensionality of the considered system
Summary
Since the seminal work of Reynolds, to whom we owe the ubiquitous Reynolds number, on transition to turbulence in pipes (Reynolds 1883) and the discovery of the famed inflection. Unsteady flows are notoriously challenging to analyse and progress in the understanding of their linear stability characteristics has been slow (Schmid & Henningson 2001) Significant steps towards this goal have been made only for time-periodic flows using energy theory (Davis & Von Kerczek 1973), non-modal stability analysis (Xu, Song & Avila 2021) and especially Floquet theory (Davis 1976). These characteristics have led to several applications ranging from control of linear instabilities (Blanchard et al 2018; Blanchard & Sapsis 2019c), prediction of dynamical events in a statistical framework (Farazmand & Sapsis 2016), the computation of sensitivities (Donello, Carpenter & Babaee 2020) to edge tracking (Beneitez et al 2020), leveraging the ability of the OTD modes to follow the linear dynamics even along a chaotic trajectory In these works, the OTD basis was used primarily as a stable and efficient numerical tool to obtain a basis of the most unstable subspace, but the structure of the OTD modes themselves has not been in the focus.
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