Fluid turbulence, the chaotic time-dependent motion of liquids and gases flowing at high speed, is of fundamental importance for engineering applications: it controls the drag on cars, aircraft and ships, is a major contributor to unwanted energy dissipation in pipelines, yet mixing of chemicals and thus clean combustion relies on it. Despite its importance and although the equations describing fluid flows are known for more than a century, our understanding of turbulence remains incomplete. As of now, we cannot derive properties of turbulent flows directly from the known flow equations. This poses a fundamental physics challenge which lead the late Nobel laureare Richard Feynman to describe turbulence as the 'most important unsolved problem of classical physics', yet it also affects applications, where the lack of a first-principle-based description of turbulence necessitates the use of approximate models with often uncontrolled errors. Consequently, controlling turbulent flows in industrial applications remains challenging and inaccurate descriptions of atmospheric and oceanic turbulence limit the reliability of weather and climate models.
Since the identification of deterministic chaos in the mid 20th century the dream to describe and eventually control fluid turbulence using dynamical systems or chaos theory concepts emerged. The idea centers on the existence of special non-chaotic but time-periodic solutions of the nonlinear flow equations. These so-called periodic orbits are dynamically unstable and thus not directly observed in a turbulent flow, but the turbulent dynamics represented by a chaotic trajectory in the flow's state space always closely shadows the unstable periodic orbits. Consequently, properties of turbulence are controlled by these special unstable periodic orbit (UPO) solutions with ergodic averages for the turbulent properties we wish to describe expressed as weighted averages over the UPOs.
While these special UPO solutions of the flow equations carry the promise to finally yield what E. Hopf, one of the founding fathers of ergodic theory, envisioned in 1948, namely a "rational theory of statistical hydrodynamics where [...] properties of turbulent flow can be mathematically deduced from the fundamental equations of hydrodynamics", such a first-principle based description of turbulence remains elusive. The major road block is that we are missing robust methodologies to computationally identify UPOs and thus we cannot find sufficiently large sets of the special time-periodic, dynamically unstable, non-chaotic solutions of the flow equations to realize a quantitative description of turbulence.
