From massive hurricanes swirling across the Earth to an aeroplane flying through turbulent air or even the way cream mixes into your coffee, these everyday phenomena are governed by turbulence. It’s a chaotic, unpredictable dance of fluids that researchers have been trying to understand for centuries. More recently, as climate change brings about more extreme weather events and as engineers push the boundaries of aircraft design, understanding turbulence has become even more urgent.
A team of researchers from the Indian Institute of Science Education and Research (IISER) Kolkata, International Centre for Theoretical Sciences (ICTS), TIFR, and Johns Hopkins University, USA, has made a significant leap forward, using a new approach to study this chaotic behaviour. Their study reveals how tiny, seemingly insignificant fluctuations play an important role in shaping the wild, swirling world around us.
For decades, scientists have known that turbulence is a chaotic system, meaning even a minuscule change at the start can lead to vastly different outcomes later on—the famous butterfly effect. The degree of this chaos is often measured by a quantity called the Lyapunov exponent, which indicates how rapidly minor differences in a fluid flow grow over time. The faster these differences grow, the more chaotic the system.
In 1963, mathematician Edward Norton Lorenz published a highly cited, seminal paper called Deterministic Nonperiodic Flow in which he wrote "...whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world." |
Early theories, such as those based on the work of the scientist Kolmogorov, suggested a straightforward relationship. It predicted that chaos would scale with the square root of the Reynolds number, a value that indicates the degree of turbulence in a fluid flow. It measures the degree of turbulence in the flow; a high Reynolds number indicates a highly turbulent flow, such as a fast-moving river. This older idea implied uniform chaos.
However, real-world observations and computer simulations have consistently shown that turbulence is far more complex, exhibiting a phenomenon known as intermittency. This means that the chaos isn't spread evenly throughout the fluid; instead, there are sudden, intense bursts of activity mixed with calmer periods.
The new research tackles this long-standing puzzle by introducing a new tool called decorrelators. They are a way to track how differences between two almost identical fluid flows evolve. Decorrelators allow scientists to precisely measure how that tiny difference grows and spreads throughout the fluid. This concept, initially developed to understand chaos in complex systems such as many-body physics, where many particles interact, was adopted by researchers for the first time to study driven-dissipative systems, including turbulence. These are systems that are constantly being pushed by external forces, such as wind on water, and lose energy due to friction.
To test their ideas, the team employed two primary approaches. First, they ran computer simulations of fluid flow, known as Direct Numerical Simulations (DNS), which directly solve the fundamental equations of fluid motion called the Navier-Stokes equations. These equations are the bedrock of fluid dynamics, but they are incredibly difficult to solve due to their complexity. By comparing two nearly identical fluid flows in these simulations, they could observe how minor initial differences grew.
They also employed a more straightforward, yet still capable, model known as the Gledzer-Ohkitani-Yamada (GOY) shell model. This model simplifies the Navier-Stokes equations into a series of interconnected shells, making it easier to simulate very high Reynolds numbers, which are typically impossible to reach with full DNS. The beauty of using both methods is that if both yield similar results, it strengthens the confidence in the findings.
They found that the Lyapunov exponent, the measure of chaos, scales with the Reynolds number in a very specific way, with a precise value of 0.59 ± 0.04. This is a significant departure from the older Kolmogorov prediction of 0.5 and other recent measurements. More importantly, they showed this happens due to the intermittent fluctuations in the fluid's velocity-gradient tensor, which describes how the fluid is stretching and deforming.
Essentially, the bursts of intense activity, or intermittency, are the true drivers of the chaotic scaling. The researchers analysed the interplay between two competing forces in the fluid: the strain (how the fluid deforms) and viscous dissipation (how energy is lost due to internal friction). They found that the strain term is the dominant factor driving the exponential growth of chaos and that these intermittent fluctuations profoundly influence its effect. This finding was remarkably consistent across both their complex DNS and the simpler GOY shell model, holding true over an astonishing seven decades of Reynolds numbers, showcasing the robustness of their new theory.
This work provides a precise, experimentally confirmed value for how chaos scales with the Reynolds number, an answer that has eluded scientists for years. Previous theories often relied on assumptions that didn't fully account for the complex, non-uniform nature of turbulence. By directly linking the scaling of chaos to intermittent fluctuations, the researchers have offered a microscopic explanation rooted in the Navier-Stokes equations themselves, moving beyond more generalised phenomenological models. The introduction and adaptation of "decorrelators" from many-body chaos is a novel methodological improvement, offering a new lens through which to study these complex systems.
However, the study also points to areas for future exploration. While the decorrelators provide a powerful macroscopic view, the researchers acknowledge that a deeper dive into the local characteristics of these fluctuations, how they behave at very specific points in the fluid, is still needed. For instance, the exact origins of the power-law tails in the distribution of these local fluctuations and their implications for how individual fluid particles move within a turbulent flow known as Lagrangian chaos are left for more detailed future studies. The study was also published on the arXiv open-access server and has not undergone peer review.
The research offers a deeper, more accurate understanding of turbulence. Beyond an academic exercise, it has immense practical implications. For meteorologists, better models of atmospheric turbulence can lead to more precise weather forecasts. In aerospace engineering, understanding turbulence is critical for designing more fuel-efficient aircraft, reducing drag, and improving safety during flights. For oceanographers, it can help predict the movement of ocean currents. By unravelling the hidden mechanisms that drive chaos in fluids, this research brings us closer to harnessing and predicting some of nature's most powerful and unpredictable forces, ultimately leading to a safer, more efficient world.