Into the vortex

The subject of vortices might seem esoteric. But their impact does make headlines, as seen recently in an outbreak of tornadoes, swirling vortices that killed at least 80 people across eight U.S. states in December 2021. Scientists today still don’t fully understand the dynamics of vortices, chaotic but coherent patterns common in nature that are also exemplified by hurricanes, eddies in a stream of air or water, aerodynamic drag, fuel combustion, and more.

Supercomputer simulations are helping scientists peer deeper into the mysterious characteristics of vortices and turbulence, in recent studies by Texas Tech University scientists. A possible application of their research could help improve fuel efficiency for cars and help develop energy-saving aircraft designs, and more.

Their vortex research was published January 2022 in the Annual Review of Fluid Mechanics. “One takeaway from the study is that we find that where two opposite-signed vortices come together, they will reconnect and recombine to form two new vortices, with some remaining unreconnected parts left as threads, which can further undergo successive reconnections.” said study lead author Jie Yao, a post-doctoral researcher in the Department of Mechanical Engineering at Texas Tech.

Vortex Reconnection

“Vortex reconnection, we claim, is the essence behind most turbulence cascade, fluid mixing, and aerodynamic noise generation,” said study co-author Fazle Hussain, President’s Endowed Distinguished Chair in Engineering, Science and Medicine, and Senior Adviser to the President, Texas Tech University. Hussain is Yao’s advisor and a professor in the Departments of Mechanical Engineering, Physics, Chemical Engineering, Petroleum Engineering, Internal Medicine, and Cell Physiology and Molecular Biophysics.

Hussain gave an example of vortex reconnection in the two contrails of an airplane. Under proper atmospheric conditions, the rolling twin trailing vortices reconnect into vortex rings and thence to turbulence.

“When vortices reconnect, they create two large structures plus many small-scale structures,” Hussain said. “Initially, you see some smoke in laboratory visualizations. But as the two vortices are pulled and move apart, they pull these threads, which eventually dissipate. These details came out only through numerical simulations with supercomputers.”

Supercomputers Solving Vortex Equations

For the review study, Yao and Hussain were awarded supercomputer access to the Stampede2 system at the Texas Advanced Computing Center by XSEDE, the Extreme Science and Engineering Discovery Environment, funded by the National Science Foundation (NSF). Additionally, they took advantage of XSEDE’s Extended Collaborative Support Services (ECSS) program, which provides researchers with expertise to make the most of the supercomputer time awarded.

“Through XSEDE ECSS, Manu Shantaram of the San Diego Supercomputer Center helped us analyze our code. We had a good connection and discussion with him, and he did a good job in profiling the code and finding problems, which improved its performance,” Yao said.

“We’ve benefitted greatly from XSEDE projects, and even more from TACC, whose staff helped us with technical issues and to resolve problems,” Yao added. “And TACC provides us with more than just access to Stampede2. TACC has also awarded us access to the systems Frontera and Lonestar5, in addition to the new Lonestar6.”

Yao and Hussain have harnessed considerable supercomputing power from XSEDE, TACC, and their local cluster at the Texas Tech University High Performance Computing Center (HPCC). It’s all to basically solve Navier-Stokes equations, which describe the fluid motion of air, water, and more. Their direct numerical simulations have yielded highly time-and space-resolved, accurate distributions of measures such as velocity, vorticity or fluid rotation, enstrophy – a term related to energy dissipation of a vortex, helicity, temperature, and scalar concentration.

The growth of peak vorticity and enstrophy both address a very fundamental mathematical question that’s relevant to a million-dollar question posed by the Clay Mathematics Institute, who have pledged the money for a correct solution to one of several Millennium Prize Problems.

The question has to do with the formation of a finite-time singularity (FTS) of the Navier-Stokes equations, which can be stated as the issue of whether, given at some initial instant and smooth velocity field of finite kinetic energy, a singularity of the field appears within a finite time under evolution governed by the incompressible Navier–Stokes equations.

“Direct numerical simulation (DNS) using supercomputers has also been used to study the possible formation of an FTS,” Yao said. DNS computer simulations are used in computational fluid dynamics to solve Navier-Stokes equations without using a model, a computationally expensive method. He noted that simulations cannot give clear evidence of the existence of an FTS, because the length scale of the phenomenon inevitably decreases to less than the computational grid resolution.

“In particular, we found that the maximum vorticity growth during the colliding of slender vortex rings is much smaller than that predicted by theory – precluding the possible formation of finite-time singularity for this configuration. Using DNS to detect self-similarity during the initial approach phase and then introducing an appropriate scaling analysis near the singular time may be one pathway to address this challenging question, but little progress has been made yet along this direction,” Yao said.

Vortex Review

Where supercomputers have helped make progress, said Yao, is in yielding results that have created more accurate and realistic representations of vortices covered in the review.

“We mainly reviewed recent progresses of vortex reconnection in classical viscous flows, including the physical mechanism, its relationship to turbulence cascade, the formation of a finite-time singularity, helicity dynamics and aeroacoustic noise generation,” Yao said.

In an earlier study, Yao and Hussain addressed two key underlying mechanisms in turbulent flows, turbulence cascade and vortex reconnection. “We also claim and demonstrate that the reconnection is one of the dominant pathways for the cascade of energy to the finest scales of turbulence before being converted to heat via the process of dissipation,” said Yao.

One challenge for the study of vortex reconnection in viscous flows is that reconnection is never complete. It leaves the unreconnected parts as threads, which can have rich dynamics (including mixing and turbulence cascade).

Avalanche of Vortices

For example, recent they’ve completed computer simulations of reconnection at moderate Reynolds numbers, which are the ratio of inertial to viscous forces, with higher values corresponding to more turbulent flow. The simulations show the threads can further undergo a cascade of secondary reconnections.

As the Reynolds number increases, the dynamics become even more complicated.

“The collision of the vortex tubes leads to an instantaneous generation of multiple thread dipoles. These dipoles then undergo an enormous number of reconnections, causing an avalanche of a large tangle of vortices in a turbulent cloud,” Yao said.

“Avalanche,” a term used by Yao and Hussain to explain cascade in various flow situations, “is very important,” Hussain added. “We’ve shown through computer simulation that vortices reconnect from one to two, to suddenly we have many vortices.”

“Imagine vortex threads of fuel and oxygen,” Hussain said. “And suddenly the fuel and oxygen are next to each other, their vortices reconnecting. You could have more complete combustion and burn less fuel. It can be a major breakthrough.”

He also pointed out that fuel-burning vehicles such as cars, submarines, aircraft, and rockets need to overcome the drag of the surrounding air.

“It turns out that in the US civil aviation alone, if you can improve the drag by one percent, you can save three billion dollars. We have ways to suggest that maybe we could achieve a 20-30% reduction in drag. That would be phenomenal,” Hussain said.

Wall Turbulence

Yao and Hussain also studied skin friction drag reduction of wall turbulence at supersonic speeds, in work published November 2021 in the journal Physical Review Fluids of the American Physical Society.

“Drag control in wall turbulence is another important research area in our group,” Yao said, where successful control of wall turbulence requires a thorough understanding of the underlying physics.

“In our view, turbulence is a collection of many vortices of different scales,” said Hussain. During the past several decades, a major advance in wall turbulence research according to Yao is the discovery, understanding, and documentation of organized ‘coherent structures’ such as vortices and their important roles in near-wall dynamics. Vortices basically form a self-sustaining generation cycle of wall turbulence.

“In general, interrupting any stage of this self-sustaining cycle could result in the suppression of streamwise vortex generation and hence reduction of drag – reducing fuel consumption and environmental pollution. We have studied various drag control techniques both in incompressible and compressible flows. Most important, noting that the large- and very-large scale motions become dominant at high Reynolds numbers, we have proposed the large-scale spanwise opposite wall-jet forcing control and composite control techniques,” said Yao.

Turbulent Simulations

Allocations were awarded by XSEDE on Stampede2 for the vortex and turbulence research. And the team were awarded separate allocation on TACC’s Lonestar5 system and by Texas Tech University HPCC.

Yao and Hussain are continuing their pipe flow research on TACC’s NSF-funded Frontera supercomputer, the fastest academic supercomputer in the world. The main objective of their work on Frontera is to simulate turbulent pipe flow at relatively high Reynolds numbers.

Roughly, half of the energy spent in transporting fluids through pipes, or vehicles through air and water, is dissipated by turbulence in the vicinity of walls. “Hence, a clearer understanding of the associated flow physics has a direct and substantial impact, and improved knowledge of those problems will be essential to finding scientific methods to control the flow phenomena, such as drag, and heat and mass transfer,” Yao said.

“Despite being an esoteric topic,” said Hussain, “we cannot live without turbulence. The damage by the tornados and hurricanes is real. And there are examples of mixing, entrainment, combustion, drag – all these phenomena require knowledge of details, such as what we’re doing now with pipe flow. Supercomputers are not yet big enough to simulate realistic turbulence, such as at Reynolds numbers of 10 million or more found on the wing tip of a jet in flight. It takes enormous computer resources, and we’re just beginning to scratch the surface.”

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Yao and Hussain acknowledge computing allocations from Stampede2 at TACC through XSEDE TG-CTS190038; TACC Lonestar and Frontera systems; and Texas Tech University’s HPCC. The research is funded by President’s Endowed Distinguished Chair Funds in Engineering, Science, & Medicine, TTU (B56388-T). The pipe flow simulation is partially funded by the NSF grant (2031650).