Robust methods for constructing periodic orbits in wall-bounded shear flows

  • Ashtari, Omid (EPFL)
  • Zheng, Zheng (EPFL)
  • Schneider, Tobias (EPFL)

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The spatiotemporally chaotic dynamics of a turbulent flow is underpinned by the unstable non-chaotic invariant solutions embedded in the state space of the governing equations. Extracting invariant solutions, therefore, is the key for studying turbulence in the dynamical systems framework. Invariant solutions such as equilibria, periodic orbits, etc. can be constructed by solving a minimization problem over the space-time fields with prescribed temporal behavior: minimization of a cost function that quantifies the deviation of a space-time field from being a solution to the evolution equations at each and every point in space and time. The optimization approach does not suffer from sensitivity to the initial guess and small convergence radius, which are the main drawbacks associated with the alternative Newton-based shooting methods. Despite the advantages of the optimization approach, its application to 3D wall-bounded flows remains challenging. One challenge is to deal with a very high-dimensional problem, and the other is to deal with the non-linear, non-local pressure term which is not easily accessible in the presence of walls, if possible at all. The adjoint-based minimization techniques introduced by [Farazmand 2016] and [Azimi et al. 2022] scale linearly with the size of the problem, thus allow us to apply them to very high-dimensional problems. However, the application of these minimization techniques has been limited to the 2D Kolmogorov flow where the doubly periodic domain enables us to handle the pressure term ([Farazmand 2016] and [Parker & Schneider 2022]). We propose a Jacobian-free algorithm based on these adjoint-based methods for constructing invariant solutions of wall-bounded fluid flows without requiring to construct the pressure field explicitly. We demonstrate the feasibility of the algorithm by constructing equilibria and periodic orbits in plane Couette flow and inclined layer convection (see Figure 1). We also propose a data-driven procedure based on dynamic mode decomposition for accelerating the convergence of the adjoint-descent algorithm.