Search for Unstable Relative Periodic Orbits in Plane Poiseuille Flow using Symmetry-Reduced Dynamic Mode Decomposition

  • Engel, Matthias (University of Edinburgh)
  • Ashtari, Omid (Ecole Polytechnique Federale de Lausanne)
  • Schneider, Tobias (Ecole Polytechnique Federale de Lausanne)
  • Linkmann, Moritz (University of Edinburgh)

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Within the dynamical systems approach to turbulence, state-space structures such as unstable periodic orbits (UPOs) embedded in the chaotic attractor are known to carry information about the dynamics of the system and therefore provide a tool for its description. UPOs are usually found by Newton searches, and constructing good initial data is challenging. A commonly used technique to generate such initial data involves detecting recurrence events by comparing future with past flow states using their L_2-distance (Cvitanovic et al. 2010). A drawback of this method is the need for the trajectory to shadow the UPO for at least one of its cycles, which becomes less likely for higher Reynolds numbers. Furthermore, one only obtains local-in-time information. A method that bypasses both issues is based on dynamic mode decomposition (DMD), where initial guesses are constructed using a few dynamic modes corresponding to dominant frequencies identified from the data (Page et al. 2020). However, in the presence of continuous symmetries, DMD fails to provide accurate approximations of the dynamics. To address this, we combine symmetry-reduced DMD (SRDMD), an approach recently developed by Marensi et al. (2023) with sparsity promotion (Jovanovic et al. 2012). In doing so, we construct optimal low-dimensional representations of the data that are global objects, and each instant in time can be chosen as an initial guess. We apply the method to data obtained by direct numerical simulation of 3d plane Poiseuille flow at friction Reynolds number Re_tau=50, taking a shift symmetry in streamwise direction explicitly into account. We demonstrate, that the obtained unstable relative periodic orbits (URPOs) cover relevant regions of the systems state space, suggesting their importance for a possible description of the flow. Furthermore, we emphasise the advantage of this approach compared to the recurrent flow analysis, since (SR)DMD provides a set of initial guesses contained in a reconstructed trajectory and therefore, enables multi-shooting methods to operate on a global object.