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New dynamical framework for Turbulence

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Nearly all fluid flows are turbulent, exhibiting diverse spatial and temporal structures. Turbulence is chaotic, where small external disturbances can lead to remarkably different behavior as time evolves. Despite these properties, Turbulence can exhibit flow patterns that persist for substantial durations of time, known as coherent structures.

Scientists and engineers have puzzled at ways to predict and alter turbulent fluid flows, and it has long remained one of the most challenging problems in science and engineering.

Physicists from the Georgia Institute of Technology have developed a new method of detecting when Turbulence resembles these coherent flow structures. Using this method, they 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 precomputed once and for all for a specific geometry.

Roman Grigoriev, School of Physics, Georgia Institute of Technology, Atlanta, said, “For nearly a century, Turbulence has been described statistically as a random process. Our results provide the first experimental illustration that, on suitably short time scales, the dynamics of Turbulence is deterministic — and connects it to the underlying deterministic governing equations.”

“Quantitatively predicting the evolution of turbulent flows — and, in fact, almost any of their properties — is rather difficult. Numerical simulation is the only reliable existing prediction approach. But it can be costly. The goal of our research was to make prediction less costly.”

 tracking the motion of millions of suspended fluorescent particles
The setup allowed the researchers to reconstruct the flow by tracking the motion of millions of suspended fluorescent particles. Credit: Photo: Michael Schatz

By observing weak turbulent flow- confined between two independently rotating cylinders- scientists created a new roadmap of Turbulence. This allowed scientists to compare experimental observations uniquely with numerically computed flows due to the absence of “end effects” in more familiar geometries, such as flow down a pipe.

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The experiment used transparent walls to allow full visual access and cutting-edge flow visualization to enable the scientists to reconstruct the flow by tracking the movement of millions of suspended fluorescent particles. Simultaneously, they used advanced numerical methods to compute recurrent solutions of the partial differential equation (Navier-Stokes equation), which governs fluid flows under conditions identical to the experiment.

As mentioned above, turbulent fluid flows show coherent structures. By analyzing their experimental and numerical data, the scientists discovered that these flow patterns and their evolution resemble those described by the special solutions they computed.

These special solutions are recurrent and unstable, describing repeating flow patterns over short intervals. Turbulence follows one solution after another, explaining how and when patterns can appear.

schematic of the physicists' research
A schematic of the physicists’ research. Credit: Michael Schatz, Roman Grigoriev.

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

“We then compared turbulent flows in experiment and direct numerical simulations with these recurrent solutions and found Turbulence to closely follow (track) one recurrent solution after another, for as long as turbulent flow persisted. Such qualitative behaviors were 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 chaotic motion tracking recurrent solutions observed in turbulent flows. The dynamics of turbulent flows are, of course, far more complicated due to the quasi-periodic nature of recurrent solutions.”

“Using this method, we conclusively showed that these structures well capture the organization of Turbulence in space and time. These results lay the foundation for representing Turbulence in terms of coherent structures and leveraging their persistence in time to overcome the devastating effects of chaos on our ability to predict, control, and engineer fluid flows.”

“These findings most immediately impact the community of physicists, mathematicians, and engineers who are still trying to understand fluid turbulence, which remains “perhaps the greatest unsolved problem in all science.”

“This work builds and expands on previous work on fluid Turbulence by the same group, some of which was reported at Georgia Tech in 2017. Unlike the work discussed in that publication, which focused on idealized two-dimensional fluid flows, present research addresses the practically important and more complicated three-dimensional flows.”

“Ultimately, the study lays a mathematical foundation for fluid turbulence which is dynamical, rather than statistical, in nature — and hence has the capability to make quantitative predictions, which are crucial for various applications.”

Journal Reference:

  1. Christopher J. Crowley et al. Turbulence tracks recurrent solutions. Proceedings of the National Academy of Sciences. DOI: 10.1073/pnas.2120665119
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