From the gentle swirl of milk in your coffee to the fierce winds of a typhoon, rotating turbulent flows are omnipresent yet scientifically intricate. The ability to describe, model, and predict these flows holds significant implications across various fields, including weather forecasting and the study of planet formation in the accretion disks of nascent stars.
At the heart of turbulence research are two pivotal formulations: Kolmogorov’s universal framework for small-scale turbulence, which describes how energy propagates and dissipates through increasingly smaller eddies, and Taylor-Couette (TC) flows, known for their simplicity in creation but complexity in behavior. These flows set the benchmark for studying the fundamental characteristics of complex flows.
For decades, a contradiction between these powerful formulations has puzzled scientists. Despite extensive experimental research and its universal application to almost all turbulent flows, Kolmogorov’s framework seemed inapplicable to turbulent TC flows. However, researchers at the Okinawa Institute of Science and Technology (OIST) have resolved this discrepancy. After nine years of developing a world-class TC setup, they have demonstrated that Kolmogorov’s framework does indeed apply universally to the small scales of turbulent TC flows, as predicted. Their findings are 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, which occur in the closed space between two independently rotating cylinders, are straightforward to create but exhibit a wide range of turbulent behaviors. These flows lead to the formation of rotating, turbulent vortices known as Taylor rolls, akin to the vertical swirling currents in a typhoon. The analysis of these rolls has helped establish several core assumptions in fluid dynamics.
In 1941, mathematician Andrey Kolmogorov introduced an elegant formulation on the complexity of turbulent fluids, describing it as 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 Kolmogorov’s celebrated -5/3rd law being found universal across most turbulent flows, TC flows seemed to evade his framework. Numerous experiments over the years failed to fit the small-scale universality that the -5/3rd law predicts.
Resolving the Inconsistency with Data Collapse
The inconsistency has long troubled Prof. Chakraborty and other physicists. As he puts it, “how can Kolmogorov’s power law be universal if it doesn’t apply to one of the most important flow regimes in fluid mechanics?” This challenge led to the development of a new experimental setup at OIST, a project that took nine years due to the complexities involved. The setup required precise sensors within a rapidly spinning cylinder, 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 from the inertial range to 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). By applying this less-explored aspect of Kolmogorov’s framework, the team successfully demonstrated the universality Kolmogorov predicted. “Rescaling the measurements by the general theory yielded the universality that Kolmogorov predicted. The framework holds,” Dr. Barros adds.
Implications for Future Research
This elegant solution to the inconsistency in Kolmogorov’s theory unlocks the potential of turbulent TC flows as powerful tools for studying theoretical and applied fluid mechanics, especially with the new OIST-TC setup. Prof. Chakraborty summarizes: “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 breakthrough not only resolves a longstanding scientific contradiction but also paves the way for more precise and comprehensive studies in fluid dynamics. As researchers continue to explore the vast applications of this discovery, the potential for advancements in various fields, from meteorology to astrophysics, seems boundless.
