New technique provides evidence for century-old theory of turbulence

Stirring a little cream into a cup of coffee seems like a simple act, but as the two liquids mix together, things get very, very complicated: Big swirls spin off smaller swirls, which produce their own tiny swirls—and so on, in a complicated and chaotic dance that is a visual representation of turbulence.

Turbulence, the chaotic changes in flow velocity and pressure, is one of the most difficult aspects of fluid dynamics to characterize. About a century ago, researchers hypothesized that turbulence involves self-similar structures—in this case the swirls—repeated at various scales. But finding quantitative evidence of that has been difficult.

Now, a new approach by University of Wisconsin engineers is providing direct evidence of that hypothesis. They have detailed their data-driven wavelet decomposition method in a paper in the journal Proceedings of the National Academy of Sciences.

“In turbulent flows in nature, the size of the biggest scale compared to the smallest scale is an enormous ratio, and the entire range of scales is present at once,” says Daniel Floryan, a postdoctoral research associate in the lab of Michael Graham, a professor of chemical and biological engineering at UW-Madison. “That’s one of the difficulties of trying to tackle this problem.”

The new computational method developed by Floryan and Graham extracts these localized multiscale features from a dataset. For instance, images of the surface of Jupiter reveal turbulent swirls of all different sizes throughout the gas-giant’s atmosphere. The new method works from the bottom up, first learning how the very smallest swirls are structured and simultaneously extracting them from the data, then moving on to the next largest swirls, all the way up to the largest swirl patterns. Researchers then can compare and contrast these structures of various sizes, which are represented by wavelets learned from the data.

In the paper, the team applied the technique to a dataset of homogeneous isotropic turbulence, available through the Johns Hopkins Turbulence Database. The decomposition technique was able to pull out self-similar features in the isotropic turbulence data at various scales, showing that many of the features were identical, varying only in size. “The main thing we see is that at the intermediate range, the features are exact copies of each other,” says Graham. “The data is giving us direct evidence of self-similarity across the intermediate range of scales—providing direct evidence for the century-old picture of turbulence.”

Graham and Floryan hope to use the technique as a starting point to explore other turbulent flow data and other complex multiscale systems. With a few tweaks, it could be used to understand hierarchical structure in systems like biological tissue, active matter (which includes things like herds of animals or groups of cells), ocean dynamics and weather, computer and social networks, and image processing.

“There are a lot of problems where we have multiple scales present and features that are spatially isolated,” says Floryan. “This is a way to discover and extract these structures in an automated way from the data.”

Michael Graham is the Steenbock Professor of Engineering and Harvey D. Spangler Professor in chemical and biological engineering at the University of Wisconsin-Madison. This work was supported by AFOSR grant FA9550-18-0174 and ONR grant N00014-18-1-2865 (Vannevar Bush Faculty Fellowship).