Periodic Reporting for period 2 - PERTURB (Using periodic orbits to quantitatively describe and control 3D fluid turbulence.)
Reporting period: 2022-07-01 to 2023-12-31
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.
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.
To open avenues towards a quantitative description of turbulence in terms of unstable time-periodic solutions of the flow equations, we develop novel robust computational methods for identifying these UPOs. A UPO is a trajectory that (a) satisfies the flow equations and (b) is time-periodic and thus closes on itself. Standard methods for identifying UPOs start with solutions of the flow equations and consider an initial value problem for which the initial condition is varied until the trajectory also closes on itself. Matrix-free Newton methods allow this shooting method to be applied to very high-dimensional problems including discretized 3D Navier-Stokes problems but very small convergence radii make it practically impossible to find many UPOs.
Consequently, we follow an alternative approach, reversing the order in which the two properties of an UPO are enforced. That means, we start with time-periodic loops in state space and computationally deform those until the loop becomes an integral curve of the vector field induced by the flow equations. Technically, the novel loop convergence methods are based on formulating a minimization problem in the space of loops and solving the resulting variational problem using adjoint methods to circumvent the construction of Jacobian matrices. This results in very robust convergence algorithms that can be applied to very high dimensional problems including 3D fluid flows.
We have already demonstrated the superior performance of newly developed loop convergence algorithms for three systems: (a) the Kuramoto-Sivashinsky equations, a 1-dimensional nonlinear PDE used as sandbox model; (b) 2D incompressible Navier-Stokes flow with succesful handling of the nonlocal pressure and - very recently - (c) 3D channel flow described by the incompressible Navier-Stokes equations with no-slip boundary conditions on the walls.
Together with data-driven acceleration techniques and machine-assisted generation of initial guesses the vastly more robust convergence properties of the developed adjoint-based variational tools will allow to compute large sets of UPOs and thereby enable us to explore a quantitative description of turbulence in terms of these special non-chaotic solutions of the underlying flow equations.
Consequently, we follow an alternative approach, reversing the order in which the two properties of an UPO are enforced. That means, we start with time-periodic loops in state space and computationally deform those until the loop becomes an integral curve of the vector field induced by the flow equations. Technically, the novel loop convergence methods are based on formulating a minimization problem in the space of loops and solving the resulting variational problem using adjoint methods to circumvent the construction of Jacobian matrices. This results in very robust convergence algorithms that can be applied to very high dimensional problems including 3D fluid flows.
We have already demonstrated the superior performance of newly developed loop convergence algorithms for three systems: (a) the Kuramoto-Sivashinsky equations, a 1-dimensional nonlinear PDE used as sandbox model; (b) 2D incompressible Navier-Stokes flow with succesful handling of the nonlocal pressure and - very recently - (c) 3D channel flow described by the incompressible Navier-Stokes equations with no-slip boundary conditions on the walls.
Together with data-driven acceleration techniques and machine-assisted generation of initial guesses the vastly more robust convergence properties of the developed adjoint-based variational tools will allow to compute large sets of UPOs and thereby enable us to explore a quantitative description of turbulence in terms of these special non-chaotic solutions of the underlying flow equations.
The developed loop convergence methods complement and appear to vastly outperform standard methods that essentially all previous work within the community studying fluid turbulence using dynamical systems methods has been based on. Consequently, a much larger number of UPOs should be accessible. To make the methods available to the community, we plan to include the variational tools in our own open-source software package employed by a growing number of research groups worldwide. Together with an envisioned public database for UPOs we will provide the framework for collecting a sufficiently large set of UPOs in a crowd-sourcing fashion. Thereby, we plan to test the feasibility of a quantitative description of turbulence in geometrically simple geometries such as parallel channels and - at least initially - at transitional Reynolds numbers that is directly derived from the fundamental hydrodynamic equations.
While the computational tools are developed within the context of shear flow turbulence, we expect the methods to also be relevant in other fields ranging from active media to nonlinear optics and beyond.
While the computational tools are developed within the context of shear flow turbulence, we expect the methods to also be relevant in other fields ranging from active media to nonlinear optics and beyond.