Using periodic orbits to quantitatively describe and control 3d fluid turbulence.
Fluid turbulence is of key importance in engineering: it controls the drag on aircraft, is a major contributor to unwanted energy dissipation in pipelines, yet mixing of chemicals relies on it. Despite its importance, our understanding of turbulence is incomplete. Controlling turbulent flows, which would lead to significant efficiencies for industrial applications, remains a challenge. 

The recent identification of unstable non-chaotic solutions of the Navier-Stokes equations suggests a promising framework to study the phenomenon. Here turbulence is viewed as a chaotic walk through a forest of exact solutions in the infinite-dimensional state space of the flow equations. While this dynamical systems approach helps rationalizing features of the transition to turbulence, it has so far failed to deliver on the promise it carried since the identification of deterministic chaos in the mid 20th century: To provide a predictive description of turbulence in terms of exact solutions and to act as a rational basis for controlling flows. 

The major road block to fulfilling the promise is that we are missing tools to identify enough dynamically relevant exact solutions. These are time-periodic non-chaotic solutions that allow us to express statistical properties of turbulence as a weighted average over periodic orbits. Owing to the exponential error amplification in a chaotic system, periodic obits for 3D flows have been almost inaccessible. Instead, research has focused on isolated steady solutions that resemble features of the flow but are dynamically less informative. 

We will remove the road block and construct extensive libraries of periodic orbits for two canonical 3D flows, turbulent convection and channel flow. By combining variational methods with machine learning tools we will automatically compute periodic orbits of the 3D Navier-Stokes equations. Using periodic orbit theory, we will describe and control flow properties including heat transport and turbulent drag.