Computing Simple Invariant Solutions in Chaotic Fluid Flows

Invariant solutions of the Navier–Stokes equations, such as fixed points, traveling waves, and periodic orbits, provide a dynamical systems framework for turbulence. Although dynamically unstable, these solutions organize chaotic dynamics and capture coherent structures observed in experiments and simulations. Over the past three decades, substantial progress has been made in computing invariant solutions, initially focusing on isolated fixed points and traveling waves and, more recently, shifting toward libraries of periodic orbits that offer a more complete description of the dynamics. Computing these solutions is challenging because of their instability, the high dimensionality arising from spatial discretization, and the increasing complexity in more turbulent regimes. This review provides an overview of the numerical methods developed to address these challenges. We discuss matrix-free Newton–Krylov approaches, more recent variational methods designed to improve robustness in the computation of unstable periodic orbits, and emerging strategies that combine equation-based algorithms with data-driven techniques to generate improved initial guesses. Together, these developments provide increasingly powerful tools for exploring the dynamical structure of turbulence.