From the gentle swirl of milk in a coffee cup to the fierce winds of a typhoon, rotating turbulent flows are omnipresent in our world. Despite their ubiquity, these spinning currents pose significant scientific challenges. The ability to describe, model, and predict turbulent flows holds immense importance across diverse fields, including weather forecasting and the study of planetary formation in the accretion disks of nascent stars.
Central to the study of turbulence are two key formulations: Kolmogorov’s universal framework for small-scale turbulence, which explains how energy cascades and dissipates through progressively smaller eddies, and Taylor-Couette (TC) flows, which, despite their simplicity, display highly complex behaviors. These flows have become a benchmark for understanding the fundamental characteristics of complex fluid dynamics.
Resolving a Decades-Old Contradiction
For decades, a significant contradiction between these formulations has puzzled scientists. While Kolmogorov’s framework is deemed universal for almost all turbulent flows, it seemingly failed to apply to turbulent TC flows. However, researchers at the Okinawa Institute of Science and Technology (OIST) have recently resolved this contradiction. After nine years of developing a state-of-the-art TC setup, they demonstrated that Kolmogorov’s framework does, in fact, universally apply to the small scales of turbulent TC flows, as originally predicted. Their groundbreaking findings have been published in Science Advances.
“The problem has long stood out like a sore thumb in the field,” said 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.”
The Complexity of Taylor-Couette Flows
Taylor-Couette flows are generated in closed systems between two independently rotating cylinders. Despite their simple creation, these flows exhibit a broad spectrum of turbulent behaviors, including the formation of rotating, turbulent vortices known as Taylor rolls. These vertical swirling currents resemble the air currents in a typhoon and have been instrumental in establishing several core assumptions in fluid dynamics.
In 1941, mathematician Andrey Kolmogorov introduced an elegant formulation on the complexity of 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 universality of Kolmogorov’s celebrated -5/3rd law across virtually all turbulent flows, TC flows have consistently defied this framework. Numerous experiments over the decades have failed to align with the small-scale universality predicted by the -5/3rd law.
Universality Regained: A New Experimental Approach
The inconsistency has long troubled Prof. Chakraborty and his peers. “How can Kolmogorov’s power law be universal if it doesn’t apply to one of the most important flow regimes in fluid mechanics?” he pondered. This question spurred the development of a new experimental setup at OIST, a feat of engineering that took nine years to perfect. The challenge lay in housing precise sensors within a cylinder spinning at thousands of rpm, surrounded by liquid cooled to a constant temperature, all within another spinning cylinder capable of producing turbulent flows at Reynolds numbers up to 106, among the highest achieved globally.
“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 that dissipate energy into heat. At these scales, Kolmogorov predicted that when accounting for dissipative effects, the rescaled energy spectra would collapse onto a single, universal curve F(kη). This broader approach revealed the universality Kolmogorov had predicted, confirming the validity of his framework.
Implications and Future Prospects
This resolution of the universality inconsistency in Kolmogorov’s theory unlocks new potential for studying turbulent TC flows, both theoretically and in applied fluid mechanics, particularly with the advanced OIST-TC setup. Prof. Chakraborty emphasizes the significance of this breakthrough: “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.”
This development not only enhances our understanding of fluid dynamics but also sets the stage for further explorations into the complex phenomena of turbulent flows, potentially impacting a wide range of scientific and engineering fields.