Turbulence plays a key role in our daily lives, making airplane journeys bumpy, affecting weather and climate, limiting the fuel efficiency of the cars we drive, and impacting clean energy technologies. Yet scientists and engineers have struggled with how to predict and modify turbulent fluid flows, and it has long remained one of the most difficult problems in science and engineering.

Now physicists at the Georgia Institute of Technology have demonstrated – numerically and experimentally – that turbulence can be understood and quantified using a relatively small set of special solutions to the governing equations of fluid dynamics that can be precalculated for a particular geometry, once and for all.

“For nearly a century, turbulence has been statistically described as a random process,” said Roman Grigoriev. “Our results provide the first experimental illustration that, on sufficiently short timescales, turbulence dynamics are deterministic and relate it to underlying deterministic governing equations.”

The findings were published in Proceedings of the National Academy of Sciences on Aug. 19, 2022. The team of researchers was led by Grigoriev and Michael Schatz, professors at Georgia Tech’s School of Physics who have collaborated on various research projects over the past two decades.

Schatz and Grigoriev were joined in the study by School of Physics graduate students Chris Crowley, Joshua Pughe-Sanford and Wesley Toler, as well as Michael Krygier, a postdoctoral researcher at Sandia National Laboratories, who developed the numerical solvers of studying as a graduate student at Georgia Tech.

A new “roadmap” for turbulence research

Quantitatively predicting the evolution of turbulent flows – and indeed almost all of their properties – is rather difficult. “Numerical simulation is the only reliable prediction approach that exists,” Grigoriev said. “But that can be terribly expensive. The goal of our research was to make prediction cheaper.”

The researchers created a new turbulence ‘roadmap’ by examining a weak turbulent flow confined between two independently rotating cylinders, giving the team a unique way to compare experimental observations with numerically calculated flows, due to the absence of “end effects” that are present in more familiar geometries, such as flow in a pipe.

“Turbulence can be thought of as a car following a sequence of roads,” Grigoriev said. “Perhaps an even better analogy is a train, which not only follows a railway on a prescribed schedule, but is also the same shape as the railway it follows.”

The experiment featured transparent walls to allow full visual access, and it used state-of-the-art flow visualization to allow researchers to reconstruct flow by tracking the movement of millions of suspended fluorescent particles. In parallel, advanced numerical methods were used to calculate recurrent solutions of the partial differential equation (Navier-Stokes equation), governing fluid flows under conditions corresponding exactly to the experiment.

It is well known that turbulent fluid flows exhibit a repertoire of patterns – called “coherent structures” in the field – which have a well-defined spatial profile but appear and disappear in a seemingly random fashion. By analyzing their experimental and numerical data, the researchers found that these flow patterns and their evolution resemble those described by the special solutions they calculated. These special solutions are both recurrent and unstable, meaning that they describe repetitive flow patterns over short time intervals. Turbulence follows one of these solutions after another, which explains which patterns can appear and in what order.

Recurrent solutions, two frequencies

“All the recurrent solutions we found in this geometry turned out to be quasi-periodic, that is, characterized by two different frequencies,” Grigoriev said. One frequency describes the overall rotation of the flow pattern around the axis of flow symmetry, while the other describes changes in the shape of the flow pattern in a co-rotating frame of reference with the pattern. The corresponding flows repeat periodically in these co-rotating frames.

“We then compared turbulent flows in direct numerical experiments and simulations with these recurrent solutions and found that turbulence closely followed (followed) one recurrent solution after another, as long as the turbulent flow persisted,” said Grigoriev. “Such qualitative behaviors have been predicted for low-dimensional chaotic systems, such as the famous Lorenz model, derived six decades ago as a greatly simplified model of the atmosphere.”

The work represents the first experimental observation of recurrent chaotic motion tracking solutions actually observed in turbulent flows. “The dynamics of turbulent flows are, of course, much more complicated due to the quasi-periodic nature of recurrent solutions,” Grigoriev added.

“Using this method, we have shown conclusively that the organization of turbulence in both space and time is well captured by these structures,” the researchers said. “These findings lay the groundwork for representing turbulence in terms of coherent structures and leveraging their persistence over time to overcome the devastating effects of chaos on our ability to predict, control, and design fluid flows.”

A new dynamic basis for 3D fluid flows

These findings have an immediate impact on the community of physicists, mathematicians and engineers still trying to understand fluid turbulence, which remains “perhaps the greatest unsolved problem in all of science,” Grigoriev said.

“This work builds on and expands on previous work on fluid turbulence by the same group, some of which was reported to Georgia Tech in 2017,” he added. “Unlike the work discussed in this publication, which focused on idealized two-dimensional fluid flows, current research is focused on practically large and more complicated three-dimensional flows.”

Ultimately, the team’s study establishes a mathematical basis for fluid turbulence that is dynamic, rather than statistical, in nature, and therefore has the ability to make quantitative predictions, which are crucial for a variety of apps.

“It can give us the ability to significantly improve the accuracy of weather forecasts and, more specifically, to enable prediction of extreme events such as hurricanes and tornadoes,” Grigoriev said. “The dynamic framework is also critical to our ability to design flows with desired properties, for example, reduced drag around vehicles to improve fuel efficiency, or improved mass transport to help remove more carbon dioxide. of the atmosphere in the emerging direct air capture industry.”