Turbulence plays a major role in our daily lives, leading to bumpy rides, affecting weather and climate, reducing the fuel efficiency of the cars we drive, and affecting clean energy technologies. However, scientists and engineers have puzzled over methods for predicting and changing turbulent fluid flows, and it has long remained one of the most challenging problems in science and engineering.
Now, physicists from Georgia Institute of Technology have demonstrated — numerically and experimentally — that turbulence can be understood and quantified with the help of a relatively small set of special solutions to the equations governing fluid dynamics that can be pre-computed for a given geometry, once and for all.
“Nearly a century ago, perturbation was described statistically as a stochastic process,” Roman Grigoriev said. “Our results provide the first empirical demonstration, on appropriately short time scales, that perturbation dynamics is deterministic—and relate it to the underlying deterministic equations.”
The results have been published in Proceedings of the National Academy of Sciences on August 19, 2022. The team of researchers was led by Grigoriev and Michael Schatz, two professors in the School of Physics at Georgia Tech who have collaborated on various research projects over the past two decades.
Schatz and Grigoriev were joined in the study by graduate students in the School of Physics Chris Crowley, Joshua Pugh-Sanford, and Wesley Tueller, along with Michael Krieger, a postdoctoral scientist at Sandia National Laboratories, who developed numerical solutions for the study as a graduate student at Georgia Tech.
New ‘road map’ for disorders research
Quantitatively predicting the evolution of turbulent flows – and indeed, almost any of their properties – is rather difficult. “Numerical simulation is the only reliable current prediction method,” said Grigoriev. “But it can be very expensive. The goal of our research was to make prediction less expensive.”
Researchers have created a new ‘road map’ for the disorder by looking at vulnerability turbulent flow which were sandwiched between two independently rotating cylinders—giving the team a unique way to compare experimental observations with numerically calculated flows, given the absence of the “end effects” found in more common geometries, such as down-tube flow.
“Turbulence can be thought of as a car following a series of roads,” Grigoriev said. A better analogy is perhaps a train, which does not follow a specific railway only timetable But it also has the same shape as the railroad it follows.”
Featuring transparent walls to allow full visual access, the experiment used state-of-the-art flow visualization to allow researchers to reconstruct the flow by tracking the movement of millions of suspended fluorescent particles. In parallel, advanced numerical methods were used to calculate repeated solutions of the partial differential equation (the Navier-Stokes equation), which govern fluid flows under conditions exactly identical to the experiment.
Turbulent fluid flows are known to display a repertoire of patterns – referred to as “coherent structures” in the field – that have a well-defined spatial shape but appear and disappear in an apparently random manner. By analyzing their experimental and numerical data, the researchers discovered that these flow patterns and their evolution are similar to those described in the particular solutions they calculated. These particular solutions are iterative and unstable, which means that they describe flow patterns that are repetitive over short timescales. Turbulence tracks solution by solution, explaining which patterns can emerge and in what order.
Frequently repeated solutions
“It turns out that all the recurring solutions that we found in this geometry are quasi-periodic – that is, they are characterized by two different frequencies,” said Grigoriev. One frequency described the general rotation of the flow pattern around the flow symmetry axis, while the other described changes in the shape of the flow pattern in a frame of reference that alternates with the pattern. Corresponding streams are repeated periodically in these co-rotating frames.
“We then compared the turbulent flows in the experiment and direct numerical simulations with these iterative solutions and found the turbulence to closely follow (track) one iterative solution after another, as long as the turbulent flow persists,” Grigoriev said. “Such specific behaviors of low-dimensional chaotic systems were predicted, 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 cyclonic solutions to track chaotic motion already observed in turbulent flows. “The dynamics of turbulent flows are of course much more complex due to the quasi-cyclical nature of recurring solutions,” Grigoriev added.
“Using this method, we have shown conclusively that the regulation of perturbations in both space and time is well captured by these structures,” the researchers said. “These findings lay the foundation for representing turbulence in terms of coherent structures and leveraging their stability in time to overcome the devastating effects of turbulence on our ability to predict, control, and engineer fluid flows.”
A new dynamic foundation for 3D fluid flows
These findings directly impact the community of physicists, mathematicians and engineers still trying to understand fluid turbulence, which remains “perhaps the biggest 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 were Reported on Georgia Tech in 2017He added, “In contrast to the work discussed in this publication, which focused on idealized 2D liquid flows, the current research addresses the practically important and more complex 3D flows.”
Ultimately, the team’s study lays a mathematical foundation for fluids disturbance It is dynamic, not statistical, in nature – and thus has the ability to make quantitative predictions, which are critical for a variety of applications.
“It could give us the ability to significantly improve the accuracy of weather forecasts, most notably enabling the prediction of extreme events such as hurricanes and hurricanes,” Grigoriev said. “The dynamic framing is also essential for our ability to engineer flows with desired properties, for example, reducing drag around vehicles to improve fuel efficiency, or enhancing mass transit to help remove more carbon dioxide from the atmosphere in the emerging direct air capture industry.”
Christopher J. Crowley et al., Turbulence Tracking Frequent Solutions, Proceedings of the National Academy of Sciences (2022). DOI: 10.1073/pnas.2120665119
Georgia Institute of Technology
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