The Story of Turbulence: from Worms to Exascale Simulations

“Big whirls have little whirls, which feed on their velocity,

And little whirls have lesser whirls, and so on to viscosity.” 

— Lewis F. Richardson (1881-1953), Mathematician, Physicist and meteorologist.

This simple poem beautifully explains the heart of turbulence, the process by which energy flows from large, swirling motions to smaller and smaller ones until it finally dissipates as heat. This transfer of energy, called the energy cascade, is one of the most fascinating and challenging problems in fluid mechanics. Over the decades, scientists have used Direct Numerical Simulations (DNS) to uncover the hidden structures and behaviors within turbulent flows, pushing the limits of both physics and computing.

In 1993, Jiménez et al. explored “The Structure of Intense Vorticity in Isotropic Turbulence” using a 130-million-point simulation. They discovered thin, tube-like vortices nicknamed “worms” where the vorticity was extremely strong. These “worms” scale with the Kolmogorov microscale, the smallest size of motion before viscosity dominates. Their finding showed that turbulence is not just random motion but contains organized, coherent structures that carry intense rotational energy.

A decade later, Kaneda et al. (2003) expanded this understanding with 700 million grid points in “Energy Dissipation Rate and Energy Spectrum in High Resolution DNS”. They confirmed that, at high Reynolds numbers, the mean energy dissipation rate becomes independent of viscosity, a major prediction of Kolmogorov’s theory. This work also emphasized how challenging DNS is, as increasing turbulence requires exponentially more computational resources.

Then came the era of extreme events. In 2015, Yeung et al. published “Extreme Events in Computational Turbulence”, using a staggering 5.5×10⁵ million grid points to study rare, violent fluctuations in turbulence. These extreme event bursts, where energy dissipation and vorticity exceed average values by tens of thousands, revealed that turbulence is highly intermittent and unpredictable at small scales.

Further studies, such as Iyer, Sreenivasan, and Yeung (2020) in “Scaling Exponents Saturate in Three-Dimensional Isotropic Turbulence”, examined how velocity increments deviate from self-similarity. They found that certain scaling laws “saturate” at high Reynolds numbers, hinting at the presence of vortex sheets, thin, energetic layers rather than simple eddies, making turbulence even more complex than previously thought. Most recently, in 2024, Yeung et al. achieved an extraordinary leap with “GPU-Enabled Extreme-Scale Turbulence Simulations”, running DNS on the Frontier exascale supercomputer with 35 trillion grid points. Using Fourier pseudo-spectral methods and GPU acceleration, they demonstrated how next-generation computing can simulate turbulence with unprecedented detail, a step toward truly understanding the physics behind chaotic flows.

Across all these milestones, one key principle endures: in the inertial subrange, the energy carried by eddies of diameter D follows a power law proportional to D⁵⁄³. For air, this range spans roughly from 0.1 cm to 1 km, a stunning reminder of how turbulence connects the smallest ripples to the largest atmospheric motions. From Richardson’s poetic insight to exascale simulations, the study of turbulence remains a beautiful blend of physics, mathematics, and computational power, revealing the hidden order within chaos.

Written by:

• Tanvir Ahasan

• 4th Year 1st Semester, Mechanical Engineering (Oikantik, 23rd Batch)

• Ahsanullah University of Science and Technology (AUST)

DOI links of the papers mentioned above:

• 1993: 10.1017/S0022112093002393

• 2003: 10.1063/1.1539855

• 2015: 10.1073/pnas.1517368112

• 2020: 10.1103/PhysRevFluids.5.054605 / 10.1103/PhysRevFluids.7.104605

• 2024: 10.1016/j.cpc.2024.109364

• [Image 1 Reference: Yeung, P. K., Zhai, X. M., & Sreenivasan, K. R. (2015). Extreme events in computational turbulence. Proceedings of the National Academy of Sciences, 112(41), 12633–12638. https://doi.org/10.1073/pnas.1517368112 

• [Image 2] Reference: Ghira, A. A., Elsinga, G. E., & Da Silva, C. B. (2022). Characteristics of the intense vorticity structures in isotropic turbulence at high Reynolds numbers. Physical Review Fluids, 7(10). https://doi.org/10.1103/physrevfluids.7.104605 ]