Revisiting the Turbulence Problem Using Statistical Mechanics

 
The richness of turbulence continues to pose a major challenge to theoretical physics. Due to the wide range of length and time scales intrinsic to the problem, the number of degrees of freedom in 3-D makes even detailed simulation of real turbulent flows challenging. This project uses novel statistical mechanics approaches to explore how fluids become turbulent and their properties in the strongly fluctuating turbulent state itself. Turbulence has two potentially universal scaling regimes: transitional (presumably a critical phenomenon) and high Reynolds number (presumably an asymptotic regime controlled by anomalies and associated phenomena). Thus, we approach turbulence by detailed exploration of these two scaling regimes and connect these limiting cases by advancing the understanding of turbulence-mean flow/large-scale flow interactions to form a complete narrative of turbulence from nonequilibrium statistical mechanics.

This project is loosely organized along four main directions: transitional flows, mean-flow turbulent interactions, fully developed turbulence and experimental studies. Due to a commonality of conceptual and mathematical tools, there are close synergies between these directions, with personnel actively and collaboratively engaged in multiple topics.

The transition to turbulence has now been shown in one experimental system and one computational model to be a nonequilibrium transition in the directed percolation universality class. Above this transition, real flows are dominated by the emergence or imposition of mean flows that interact with turbulence with feedback in both directions. Our goal is to construct a nonequilibrium statistical theory, including the description of rare events, that characterizes the interaction of mean flows with turbulence, both in the case where the mean flow emerges self-consistently from correlations in the turbulence and where the turbulence is driven by the mean flow. An important role is played here by extreme, rare events and multi-stability. The challenge on the experimental side is to characterize scaling laws for dissipative processes accurately and on the theoretical side, to understand how they emerge and persist up to large Reynolds numbers.

At asymptotically large Reynolds numbers, turbulence exhibits strongly singular behavior. In fact, our perspective is that it may not even be fruitful to view it as strong fluctuations about a uniform background, but rather that strong space-time localized bursts are the zeroth order solution about which one has to construct a theory. In this regime, stochasticity emerges spontaneously and fluctuations exhibit strong and anomalous behavior that calls out for renormalization group methods and related field theoretic techniques to be employed.
 

Gregory Eyink is a theoretical physicist whose work impacts numerous problems in fluid mechanics, plasma physics and foundational aspects of turbulence. His contributions use field theory, renormalization group and other statistical mechanics tools, as well as rigorous mathematical methods. Eyink revived work on Onsager’s ‘ideal turbulence’ theory, giving the first published proof of the Hölder singularity theorem and the modern formulation of the Onsager conjecture on dissipative weak solutions of the incompressible Euler equations. Eyink has extended the Onsager theory as an exact renormalization group argument to many other problems, including compressible fluid turbulence, relativistic fluid turbulence and kinetic plasma turbulence. Eyink has also worked on fluid turbulence from a field-theoretic perspective, including operator product expansion and variational formulations by effective action. Eyink has shown that the phenomenon of Lagrangian spontaneous stochasticity underlies not only the dissipative anomaly for passive scalar turbulence but also fast magnetic reconnection in astrophysical plasmas.

Gregory Falkovich works in hydrodynamics and nonequilibrium statistical physics. His works, both theoretical and experimental, span the whole field of fluid physics, from laminar flows of electrons in graphene to nonlinear waves and developed turbulence in different media. He is the author of the now classical monograph on wave turbulence and of two influential textbooks, one on turbulence, another on fluid mechanics. Among his most significant results are the discovery of an anomalous scaling in passive scalar turbulence and the conformal invariance of 2-D turbulence, prediction of the bottleneck effect, description of the role of turbulence in accelerating rain and prediction of the sling effect in droplet collisions, prediction and observation of superballistic conductance of electrons in graphene.

Nigel Goldenfeld’s research spans statistical mechanics, turbulence, nonequilibrium pattern formation and the physics of living systems. He is well-known for his work on renormalization group theory and authored what has come to be one of the standard graduate texts on the subject. He is active in developing statistical mechanics methods to problems in turbulent fluid flow, for example using renormalization group arguments to connect the scaling laws in turbulence to 90-year-old data on friction in pipe flow. This work enables the anomalous scaling exponents of turbulence to be deduced from simple measurements of pressure drop along a pipe. Goldenfeld showed that directed percolation reproduces lifetime statistics observed near the laminar-turbulent transition by Björn Hof, explained the super-exponential scaling with Reynolds number in terms of extreme value statistics, and showed how directed percolation could emerge from interactions between large-scale flows and small-scale turbulence.

Björn Hof is one of the leading experimentalists in the field of transitional turbulence, with numerous seminal discoveries that connect nonlinear dynamics to statistical mechanics of fluid flow. He made the first experimental observation of nonlinear traveling waves in pipe flow. Using computer simulation, he discovered in pipe flow the first localized periodic orbit solutions that give rise to turbulent patches with transient lifetimes. In pipe flow experiments, he showed that individual turbulent patches have finite lifetimes for all Reynolds numbers and that lifetimes as well as their characteristic splitting times scale super-exponentially with Reynolds number. This allowed him to define the critical Reynolds number for the onset of turbulence in pipe flow. Hof demonstrated experimentally and numerically that the transition to turbulence in Couette flow falls into the directed percolation universality class.