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A team of international researchers from the Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Johns Hopkins University and Duke University has discovered that a century-old theory describing turbulence in fluids also applies to a very bubbly problem: how rising bubbles stir the water around them.

Their experiments, which tracked individual bubbles and fluid particles in 3D, provide the first direct experimental evidence that so-called Kolmogorov scaling can emerge in bubble-induced turbulence. The results are in Â鶹ÒùÔºical Review Letters.

Bubble-induced turbulence happens in many places: from carbonated drinks to industrial mixing processes to the crashing ocean waves. When enough bubbles rise through a fluid, their wakes stir the surrounding liquid into a complex, turbulent motion. Understanding the rules behind this chaos is essential for improving industrial designs, climate models, and more.

Yet, a central question has long puzzled researchers: Can the mathematical theory of turbulence derived by Russian mathematician Andrey Kolmogorov in 1941—known as "K41 scaling"—apply to flows where bubbles drive the motion? Until now, conflicting experimental and computer simulation results made the answer unclear.

"We wanted to get a definitive answer by looking closely at the turbulence between and around bubbles, at very small scales," says Dr. Tian Ma, lead author of the study and physicist at the Institute of Fluid Dynamics at HZDR. To achieve that, the researchers used advanced 3D simultaneous Lagrangian tracking of both phases—a technique that allows scientists to follow both bubbles and tiny tracer particles in the surrounding water with high precision and in real time.

The experimental setup involved an 11.5 cm-wide column of water into which controlled swarms of bubbles were injected from the bottom. Four high-speed cameras recorded the action at 2,500 frames per second.

They studied four different cases, varying the bubble size and the amount of gas, to replicate realistic bubbly flows. Importantly, the bubbles that were 3 to 5 millimeters in diameter were large enough to wobble as they rose, creating strong turbulent wakes. In two of the four cases—those with moderate bubble size and density—the turbulence in the flow closely followed Kolmogorov's predictions at small scales, that is, for eddies smaller than the size of the bubbles. This marks the first time such scaling has been confirmed experimentally in the midst of a bubble swarm.

Decoding turbulence: Energy cascades from big to small

"Kolmogorov's theory is elegant. It predicts how the energy that cascades from big turbulent eddies down to smaller and smaller ones—until it's eventually dissipated through viscous effects—controls the fluctuations of the turbulent fluid motion," explains co-author Dr. Andrew Bragg from Duke University. "Finding that this theory also describes bubble-driven turbulence so well is both surprising and exciting."

The team also developed a new mathematical formula to estimate the rate at which turbulence loses energy due to viscous effects—known as the energy dissipation rate. Their formula, which only depends on two bubble-related parameters—its size and how densely packed the bubbles are—matched the experimental data remarkably well. Interestingly, they found that Kolmogorov scaling was stronger in regions outside the bubbles' direct wakes. In those wakes, the fluid is so strongly disturbed that the classic turbulent energy cascade is overpowered.

One crucial insight was that for the classic Kolmogorov "inertial range"—where his scaling laws work best—to appear clearly in bubble-induced turbulence, the bubbles would need to be significantly larger. But there's a catch: in reality, bubbles of such large sizes would break apart due to their own instability.

This means there is a fundamental limit to how well the K41 theory can apply to bubbly flows. "In a way, nature prevents us from getting perfect Kolmogorov turbulence with bubbles. But under the right conditions, we now know it gets close," says Dr. Hendrik Hessenkemper, a co-author on the study who performed the experiments.

The findings not only settle an ongoing scientific debate but could also help engineers better design bubble-based systems, from chemical reactors to wastewater treatment. And for physicists, it adds another system—bubbly flows—to the growing list of chaotic phenomena where Kolmogorov's 1941 theory proves surprisingly robust.

The researchers emphasize that their study is just the beginning. Future work could investigate how turbulence behaves with even more complex bubble shapes, bubble mixtures, or under different gravitational or fluid conditions.

"The more we understand the fundamental rules of turbulence in bubbly flows, the better we can harness them in real-world applications," says Dr. Ma. "And it's pretty amazing that a theory from over 80 years ago continues to hold up in such a bubbly environment."