Transient linear stability of pulsating Poiseuille flow using optimally time-dependent modes

The analysis of time-dependent flows is notoriously challenging for linear stability methods. The Optimally Time-Dependent (OTD) modes is a recent linear framework to construct an orthonormal basis of the instantaneously most unstable directions of the tangent space. The resulting subspace can be used to extract information about the time-asymptotic and transient stability of the trajectory. We analyse the corresponding instantaneous OTD modes for pulsating Poiseuille flow, an archetypal non-autonomous fluid system, to explore the potential of the method for the transient linear stability analysis of general time-dependent flows. The time-asymptotic results of Floquet theory are confirmed by computing the Finite-Time Lyapunov exponents for a large parameter range. The comparison between the OTD modes in the limit cycle and the eigenmodes of the Orr-Sommerfeld operator for Poiseuille flow reveals the dominant intracyclic instability mechanism corresponding to modulated Tollmien-Schlichting waves as well as the transient disappearance of these structures during part of the cycle. The full non-normal growth potential of the OTD subspace is found to be nearly identical to that of Poiseuille flow. The OTD spectrum exhibits subharmonic eigenvalue orbits that are due to the presence of exceptional points in the instantaneous Orr-Sommerfeld spectrum.