From the gentle swirl of milk in your coffee to the fierce gales of a typhoon, rotating turbulent flows are a ubiquitous yet scientifically complex phenomenon. These spinning currents, while common, pose significant challenges in terms of description, modeling, and prediction. Understanding these flows is crucial for fields ranging from weather forecasting to the study of planetary formation in the accretion disks of nascent stars.
At the heart of turbulence research are two key formulations: Kolmogorov’s universal framework for small-scale turbulence, which details how energy cascades through progressively smaller eddies, and Taylor-Couette (TC) flows, which, despite their simplicity, exhibit highly complex behaviors. These flows have long served as a benchmark for studying the fundamental characteristics of complex fluid dynamics.
For decades, a central contradiction has puzzled scientists: Kolmogorov’s framework, though universally applicable to most turbulent flows, seemed inapplicable to TC flows. However, after nine years of meticulous research at the Okinawa Institute of Science and Technology (OIST), researchers have resolved this contradiction. They have demonstrated that Kolmogorov’s framework does indeed apply to the small scales of turbulent TC flows, aligning with theoretical predictions. Their groundbreaking findings are now published in Science Advances.
“The problem has long stood out like a sore thumb in the field,” says Professor Pinaki Chakraborty of the Fluid Mechanics Unit at OIST, who led the study. “With this discrepancy solved, and with the inauguration of the OIST-TC setup, we have set a new baseline for studying these complex flows.”
Unraveling the Complexity of Taylor-Couette Flows
Taylor-Couette flows are generated between two independently rotating cylinders, creating a closed system that is both simple and complex. These flows give rise to rotating, turbulent vortices known as Taylor rolls, analogous to the swirling currents within a horizontally rotating typhoon. The analysis of these flows has informed several core assumptions in fluid dynamics.
In 1941, mathematician Andrey Kolmogorov introduced an elegant formulation on turbulent fluids, describing an idealized energy cascade. “If you stir a pool of water with a big spoon,” explains Prof. Chakraborty, “you are adding energy to the water as movement in the form of a large vortex. This vortex splits into smaller and smaller eddies, until finally dissipating as heat. While easy to observe, it was extremely difficult to describe this cascade mathematically – until Kolmogorov.”
Despite the universal applicability of Kolmogorov’s celebrated -5/3rd law across most turbulent flows, TC flows have historically defied this framework. Numerous experiments over the years have failed to align with the small-scale universality predicted by this law.
Reestablishing Universality Through Innovative Approaches
The inconsistency between Kolmogorov’s predictions and TC flows has long troubled physicists. “How can Kolmogorov’s power law be universal if it doesn’t apply to one of the most important flow regimes in fluid mechanics?” Prof. Chakraborty pondered. This question motivated the development of a new experimental setup at OIST, a project that required nine years of engineering ingenuity to perfect.
The setup involved precise sensors within a cylinder spinning at thousands of revolutions per minute, surrounded by liquid cooled to a constant temperature and encased in another spinning cylinder. This configuration achieved turbulent flows at Reynolds numbers up to 106, among the highest in the world.
“When we analyzed the energy spectra measured through the new OIST-TC setup using the conventional approach, we indeed found that Kolmogorov’s power law does not fit. And that’s when we decided to look beyond the celebrated -5/3rd law, which only applies to the inertial range,” explains Dr. Julio Barros, first author of the paper.
The team expanded their analysis beyond the inertial range to encompass the general domain of small-scale flows, including the smallest eddies where energy dissipates into heat. Kolmogorov predicted that when accounting for dissipative effects, the rescaled energy spectra would collapse onto a single, universal curve F(kη). This broader application of Kolmogorov’s framework revealed the universality that had eluded researchers for so long.
Implications and Future Directions
This resolution of the universality inconsistency in Kolmogorov’s theory unlocks new potential for using turbulent TC flows as powerful tools in both theoretical and applied fluid mechanics. Prof. Chakraborty highlights the advantages of TC flow setups: “The beauty of TC flow setups is that they are closed systems. No pumps, no obstructions in the flow. We can study the flow of whatever liquid and additive that we desire – sediments, bubbles, polymers, and so forth. And by reconciling TC flows with Kolmogorov’s theory, we now have a solid reference point.”
The findings not only validate Kolmogorov’s framework in a previously elusive context but also pave the way for future research in fluid dynamics. The OIST-TC setup could serve as a model for similar studies worldwide, enhancing our understanding of turbulence and its myriad applications.
